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+ # Watching Too Much Television is Good: Self-Supervised Audio-Visual Representation Learning from Movies and TV Shows
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+
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+ Anonymous Author(s)
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+ Affiliation
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+ Address
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+ email
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+
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+ # Abstract
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+
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+ 1 The abundance and ease of utilizing sound, along with the fact that auditory clues
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+ 2 reveal so much about what happens in the scene, make the audio-visual space a
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+ 3 perfectly intuitive choice for self-supervised representation learning. However,
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+ 4 the current literature suggests that training on uncurated data yields considerably
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+ 5 poorer representations compared to the curated alternatives collected in supervised
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+ 6 manner, and the gap only narrows when the volume of data significantly increases.
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+ 7 Furthermore, the quality of learned representations is known to be heavily influ
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+ 8 enced by the size and taxonomy of the curated datasets used for self-supervised
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+ 9 training. This begs the question of whether we are celebrating too early on catching
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+ 10 up with supervised learning when our self-supervised efforts still rely almost exclu
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+ 11 sively on curated data. In this paper, we study the efficacy of learning from Movies
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+ 12 and TV Shows as forms of uncurated data for audio-visual self-supervised learning.
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+ 13 We demonstrate that a simple model based on contrastive learning, trained on a
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+ 14 collection of movies and TV shows, not only dramatically outperforms more com
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+ 15 plex methods which are trained on orders of magnitudes larger uncurated datasets,
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+ 16 but also performs very competitively with the state-of-the-art that learns from
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+ 17 large-scale curated data. We identify that audiovisual patterns like the appearance
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+ 18 of the main character or prominent scenes and mise-en-scène which frequently
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+ 19 occur through the whole duration of a movie, lead to an overabundance of easy
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+ 20 negative instances in the contrastive learning formulation. Capitalizing on such
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+ 21 observation, we propose a hierarchical sampling policy, which despite its simplicity,
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+ 22 effectively improves the performance, particularly when learning from TV shows
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+ 23 which naturally face less semantic diversity.
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+
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+ # 24 1 Introduction
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+
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+ 25 Recently, there has been tremendous progress in self-supervised learning from still images, where the
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+ 26 standard supervised training has been outperformed in a variety of image-related tasks [7, 8, 15, 29].
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+ 27 The appeal of detaching representation learning from human annotations is rooted not only in the
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+ 28 non-trivial challenges of scaling-up the labeling process, but also in the ill-defined task of determining
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+ 29 a proper taxonomy with generalization power and transferability. Both challenges only exacerbate as
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+ 30 we move from images to videos, where the notion of time is involved and the complexity of visual
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+ 31 concepts increases. Simply considering the number of training instances or even the cardinality of
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+ 32 the label set is not sufficient to conclude if one large-scale supervised dataset is more suitable than
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+ 33 another for transfer learning in video classification tasks [20]. That is, the abundance of attention
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+ 34 which video self-supervised learning has lately received is only to be expected. While many research
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+ 35 efforts in this area extend the contributions made initially in the image domain to the video domain,
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+ 36 others, including our work, have explored harnessing additional modalities such as audio or text for
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+ 37 multi-modal self-supervised learning [2, 3, 4, 22, 27, 31, 37, 36, 39].
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+ 38 From the current state-of-the-art one makes two major conclusions. First, the quality of learned
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+ 39 representations, evaluated by fine-tuning on downstream tasks, is heavily influenced by the size and
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+ 40 taxonomy of the pretraining datasets [2, 3, 39]. Second, an uncurated pretraining dataset yields
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+ 41 considerably poorer representations compared to a curated one and the gap only narrows when the
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+ 42 total amount of pretraining data significantly increases [3]. Curated data refers to likes of supervised
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+ 43 large-scale action recognition and audio classification datasets such as Kinetics [6], IG-Kinetics
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+ 44 [12], AudioSet [11], and YouTube-8M [1]. While the human-annotated labels are not accessed for
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+ 45 self-supervised pretraining, videos being trimmed and from a label set of limited cardinality with
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+ 46 biased sampling distribution1 implicitly acts as a sort of supervision. On the other hand, an uncurated
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+ 47 data refers to likes of IG-Random[3], simply a body of unlabeled videos collected blindly with
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+ 48 none of the aforementioned careful human-involvements. That being said, we know that something
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+ 49 as simple as having access to a clean object-centric training data, like Imagenet, can be indirectly
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+ 50 exploited by contrastive self-supervised learning in image domain to obtain additional performance
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+ 51 gain [41] on the downstream tasks which exhibit similar properties. The analogous to it of course are
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+ 52 the well trimmed closed-set curated datasets which are being extensively used in the literature for
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+ 53 video self-supervised pretraining, while downstream evaluations focus on benchmarks with similar
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+ 54 characteristics. Our work aims at comprehensively exploring the efficacy of learning from Movies
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+ 55 and TV Shows, as forms of uncurated data, for audio-visual self-supervised learning.
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+ 56 Many of us can relate to an experience in movie theaters when the sound of the engine, first perceived
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+ 57 by our left ear, is gradually heard more by the right ear as a car moves from the left side of the
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+ 58 screen to the right side. Another example is a scene in which an object, like a helicopter, approaches
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+ 59 the camera from distance and eventually flies over it. In this case, the perceived sound not only
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+ 60 changes in loudness but also transitions from front to back, in concert with the visuals, giving the
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+ 61 audience a more realistic feeling as if they are indeed positioned behind the camera. Besides, with
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+ 62 art being inherently novel, two movies even if they share genres or revolve around similar story
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+ 63 lines often deliver quite different experiences and portray distinct visuals, thanks to the extremely
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+ 64 artist-driven creative process behind such productions. We hypothesize that the aforementioned high
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+ 65 audio fidelity, and inherent semantic diversity characterize long-form content2 as potentially a very
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+ 66 rich source for self-supervised multi-modal representation learning. It is worth emphasizing that in
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+ 67 spirit of uncurated data, we not only blindly sample from a large collection of movies and TV shows
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+ 68 when constructing our pretraining dataset, but also perform ablation studies on the effect of genre
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+ 69 distribution, the closest we have to taxonomy in the curated datasets, confirming that the quality of
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+ 70 learned representations is agnostic with respect to such statistics.
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+ 71 To the best of our knowledge, we are the first to solely rely on uncurated data and study the efficacy of
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+ 72 self-supervised multi-modal representation learning from movies and TV shows. Despite meaningful
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+ 73 domain gap between our pretraining data and the space of downstream tasks, we obtain representations
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+ 74 which are very competitive with those learned from curated datasets. This is particularly important as
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+ 75 we follow a much simpler modeling approach in comparison with the state-of-the art.
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+
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+ # 76 2 Related Work
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+
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+ 77 Self-supervised learning techniques define pretext tasks, mostly inspired by the natural structures
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+ 78 in the data, in order to generate supervisory signals for training. Despite the plethora of proposed
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+ 79 pretext tasks in the literature, these approaches can be coarsely divided into two groups, namely
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+ 80 pretext learning, and pretext-invariant methods. Approaches which fall in the former bucket, usually
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+ 81 apply a form of transform, randomly drawn from a parametric family, to the input data then optimize
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+ 82 for predicting the parameters of the chosen transformation. Predicting the relative position of image
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+ 83 patches [9], solving jigsaw puzzles [33], estimating artificial rotations [13], colorization [50], context
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+ 84 encoders learned through inpainting [38], and learning by counting scale and split invariant visual
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+ 85 primitives [34], are among many methods which belong to this category. Similar techniques have
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+ 86 been extended from images to videos [10, 21, 24, 25, 30, 46, 48, 49], where in addition to the
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+ 87 spatial context, the temporal domain, and the arrow of time have been heavily exploited. In contrast,
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+ 88 pretext-invariant methods [5, 7, 8, 15, 18, 17, 29, 35, 39, 44] are built on the concept of maximizing
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+ 89 mutual information across augmented versions of a single instance, and are mostly formulated as
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+ 90 contrastive learning. In other words, a pretext is used to generate different views of a single input
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+ 91 for which the learning algorithm aims to maximize the intra-instance similarity, across variety of
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+ 92 transformations. Our work falls within this category, however we function in a multi-modal realm
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+ 93 employing both audio and video.
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+ 94 Earlier works which harnessed audio and video for representation learning, have leveraged audio
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+ 95 visual temporal synchronization [22, 36], correspondence [4], and cross-modal clustering [3, 37]. The
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+ 96 work by Patrick et al.[39] proposes a generalized data transformation in order to unify a variety of
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+ 97 audio-visual self-supervised pretext tasks through a noise contrastive formulation. This work is close
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+ 98 to ours in choice of objective function and data type, yet we employ no augmentation (except modality
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+ 99 projection in the terminology of [39]), and solely focus on capitalizing the advantages of learning
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+ 100 from long-form content. Morgado et al.[31] show that cross-modal discrimination is important for
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+ 101 learning good audio and video representations, something which was also pointed out earlier in a
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+ 102 clustering framework [3]. Beyond that, [31] generalizes the notion of instance-level positive and
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+ 103 negative examples by exploring cross-modal agreement where multiple instances are grouped together
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+ 104 as positives by measuring their similarity in both the video and audio feature spaces. While we also
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+ 105 adopt a cross-modal noise contrastive estimation loss, we stick with the vanilla version, instance-level
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+ 106 positive and negatives, and do not use any memory bank feature representations. Finally, Alayrac et
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+ 107 al.[2] recently proposed a multi-modal versatile network capable of simultaneously learning from
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+ 108 audio, video and text. Building on the intuition that different modalities are of different semantic
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+ 109 granularity, audio and video are first compared in a fine-grained space while text is compared with
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+ 110 the aforementioned modalities in a lower dimensional coarse-grained space. In our experiments, we
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+ 111 compare with a variant of [2] where only audio and video modalities are utilized.
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+
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+ # 112 3 Approach
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+
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+ 113 Notations and Architecture. Our pretraining dataset is denoted by $\mathcal { X } \ : = \ : \{ \mathcal { X } _ { n } | n \in [ 1 \cdot \cdot \cdot N ] \}$
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+ 114 where $\mathcal { X } _ { n } = \{ x _ { n , m } | m \in [ 1 \cdots M _ { n } ] \}$ contains $M _ { n }$ non-overlapping audiovisual snippets which are
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+ 115 temporally segmented from the duration of the $n ^ { t h }$ long-form content in the dataset. Each snippet
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+ 116 includes both audio and video modalities, formally $x _ { n , m } = ( a _ { n , m } , v _ { n , m } )$ , where $a _ { n , m } \in \mathbb { R } ^ { 1 \times \pmb { \tilde { P } } \times Q }$
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+ 117 and $v _ { n , m } \in \mathbb { R } ^ { 3 \times T \times H \times W }$ . $T , H$ , and $W$ denote the number of frames, height and width of the video,
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+ 118 while $P$ , and $Q$ respectively stand for the number of mel filters, and audio frames. Video and audio are
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+ 119 processed through 18-layers deep $\mathrm { R } ( 2 { + } 1 ) \mathrm { D }$ [45] and ResNet [16] architectures, respectively referred
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+ 120 to as $f : \mathbb { R } ^ { 3 } \check { \mathbb { R } } ^ { d _ { f } }$ and $\dot { \boldsymbol { g } } : \mathbb { R } ^ { 1 } \dot { } \mathbb { R } ^ { d _ { g } }$ . Inspired by [7], we use projection heads, $\boldsymbol { h } _ { f } : \mathbb { R } ^ { d _ { f } } \overset { \cdot } { } \mathbb { R } ^ { d }$ and
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+ 121 $h _ { g } : \mathbb { R } ^ { d _ { g } } \mathbb { R } ^ { d }$ , to map corresponding representations into a common $d$ -dimensional space before
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+ 122 computing the contrastive loss. The shallow architecture of $h _ { f }$ and $h _ { g }$ consists of two convolution
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+ 123 layers, separated by Batch Normalization [19] and ReLU [32], followed by global average pooling.
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+ 124 Once self-supervised pretraining finished, we discard the projection heads and fine-tune $f$ and $g$ for
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+ 125 respective downstream tasks.
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+ 126 Loss Function. With a slight abuse of notation3, $\boldsymbol { B } = \{ x _ { i } = ( a _ { i } , v _ { i } ) | i \in [ 1 \cdot \cdot \cdot B ] \}$ represents a
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+ 127 minibatch of size $B$ , where video and audio modalities associated with the $i ^ { t h }$ sample, $x _ { i }$ , are denoted
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+ 128 by $v _ { i }$ and $a _ { i }$ . We use $z _ { v } ^ { i } = h _ { f } ( f ( v _ { i } ) )$ and $z _ { a } ^ { i } = h _ { g } ( g ( a _ { i } ) )$ to represent the associated embeddings
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+ 129 generated by projection heads, and optimize the noise-contrastive loss [14] shown in 1 in order
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+ 130 to maximize the symmetric joint probability between audio and video. For the $i ^ { t h }$ element in the
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+ 131 minibatch, $( z _ { v } ^ { i } , z _ { a } ^ { i } )$ serves as the positive pair, while assuming negative pairs for both modalities,
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+ 132 $\mathcal { N } _ { i } = \{ ( z _ { v } ^ { i } , z _ { a } ^ { j } ) , ( \stackrel { \sim } { z } _ { v } ^ { j } , z _ { a } ^ { i } ) | j \in [ 1 \cdots B ] , i \neq j \}$ constitutes the set of negative pairs.
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+
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+ $$
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+ \mathcal { L } = - \sum _ { i = 1 } ^ { B } \log \left( \frac { e ^ { ( z _ { v } ^ { i } ) ^ { \top } ( z _ { a } ^ { i } ) } } { e ^ { ( z _ { v } ^ { i } ) ^ { \top } ( z _ { a } ^ { i } ) } + \displaystyle \sum _ { ( z _ { v } ^ { \prime } , z _ { a } ^ { \prime } ) \in \mathcal { N } _ { i } } e ^ { ( z _ { v } ^ { \prime } ) ^ { \top } ( z _ { a } ^ { \prime } ) } } \right)
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+ $$
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+
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+ 133 Most of the previous works [2, 31, 39] normalize the embeddings before computing the contrastive
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+ 134 loss and employ a temperature hyper-parameter, often denoted by $\tau$ as in [2, 31], to control the
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+ 35 smoothness for the distribution of pairwise similarities. In contrast, we have chosen to operate in an
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+ 36 unnormalized embedding space. Besides the obvious benefit of eliminating the need for tuning $\tau$ , we
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+ 7 empirically show that such decision does not affect the quality of the learned representations.
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+ 138 Sampling Policy. Contrastive loss function shown in Equation 1 is computed over $B$ training
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+ 139 instances, each in form of an audiovisual snippet. A naive sampling policy may ignore the fact that
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+ 140 snippets comprising the pretraining dataset are in fact temporal segments that were trimmed from
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+ 141 longer-form contents, i.e. movies and TV shows. Such an assumption treats our training data as
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+ 142 independent and identically distributed random variables from $\textstyle \bigcup _ { n = 1 } ^ { N } { \mathcal { X } } _ { n }$ , which constitutes the default
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+ 143 sampling policy that is commonly used in the general deep learning literature. However, in reality,
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+ 144 commonalities and correlations do exist along the temporal axis of a movie or TV show, things like
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+ 145 audio mastering artifacts, frequent appearance of the main character’s face and voice, thematic music,
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+ 146 repetitive scenes and mise-en-scène4, all of which contribute to breaking the previously discussed
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+ 147 i.i.d assumption. This is even more pronounced when we deal with multiple episodes of the same TV
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+ 148 show appearing in the pretraining dataset5. Note that, sampling from no video data is going to be i.i.d
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+ 149 but in this case the temporal correlations extend for much longer given our entities are movies and TV
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+ 150 shows. Thus, it is more accurate to think of $\mathcal { X }$ having multiple underlying domains, oriented towards
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+ 151 exclusive properties which different long-form contents are characterized by. We hypothesize that
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+ 152 during training, model gradually discovers such patterns of commonalities, which are not semantically
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+ 153 valuable, and latches onto those to quickly minimize Equation 1 leading to poor generalization6. The
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+ 154 reason being $B \ll N$ , hence for $n \sim \mathbb { U } ( 1 , N )$ and $m \neq m ^ { \prime }$ , $\mathsf { P } ( x _ { n , m } \in B \land x _ { n , m ^ { \prime } } \in B )$ is negligible.
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+ 155 In other words, the set of negative pairs in Equation 1 mainly includes pairs for which audio and
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+ 156 video come from two different movies or TV shows, thus due to the aforementioned artifacts behave
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+ 157 as easy negatives.
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+ 158 In order to quantitatively measure our hypothesis, we define different distributions, shown in Equation
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+ 159 2, over the space of audio-visual similarity. $S ^ { + }$ indicates the space of correct matches, i.e. where
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+ 160 audio and video correspond to the same snippet. $S ^ { - }$ indicates the space where audio and video do
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+ 161 not correspond yet belong to the same movie or TV show. Finally, $\bar { \boldsymbol { S } } ^ { \neq }$ indicates the space in which
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+ 162 audio and video are sampled from two distinct long-form content, hence naturally do not correspond.
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+
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+ $$
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+ ( z _ { v } ^ { n , m } ) ^ { \mathsf { T } } ( z _ { a } ^ { n ^ { \prime } , m ^ { \prime } } ) \sim \left\{ { \begin{array} { l l } { S ^ { + } , } & { { \mathrm { i f } } \ n = n ^ { \prime } \wedge m = m ^ { \prime } } \\ { S ^ { - } , } & { { \mathrm { i f } } \ n = n ^ { \prime } \wedge m \neq m ^ { \prime } } \\ { S ^ { \neq } , } & { { \mathrm { i f } } \ n \neq n ^ { \prime } \wedge \forall ( m , m ^ { \prime } ) } \end{array} } \right.
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+ $$
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+
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+ 163 With that, and $\mathsf { K L }$ denoting Kullback–Leibler divergence, $\mathsf { K L } ( S ^ { - } \parallel S ^ { + } )$ measures the expected
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+ 164 difference between positive and negative pairs within the same movie or TV show. Ideally, this should
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+ 165 increase as the training progresses, since the model gradually learns audio-video correspondence by
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+ 166 minimizing Equation 1. Meanwhile, the i.i.d assumption suggests $\mathsf { K L } ( \mathcal { S } ^ { - } \parallel \mathcal { S } ^ { + } ) \simeq \mathsf { K L } ( \mathcal { S } ^ { \neq } \parallel \mathcal { S } ^ { + } )$
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+ 167 and $\mathsf { K L } ( \mathcal { S } ^ { - } \parallel \mathcal { S } ^ { \neq } ) \simeq \boldsymbol { 0 }$ , yet as we empirically illustrate later, $\mathsf { K L } ( \mathcal { S } ^ { - } \parallel \mathcal { S } ^ { + } ) < \mathsf { K L } ( \mathcal { S } ^ { \neq } \parallel \mathcal { S } ^ { + } )$ and
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+ 168 ${ \mathsf { K L } } ( S ^ { - } \parallel S ^ { \neq } )$ is rather large, indicating that, upon convergence and on a held-out set, model has
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+ 169 a harder time pushing apart negative pairs when audio and video come from the same underlying
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+ 170 long-form content. Next, we explain how a simple alternative policy which samples $k$ snippets
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+ 171 from each long-form content effectively reduces both of the discrepancy measures, referring to
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+ 172 $\mathsf { K L } ( S ^ { - } \parallel S ^ { \neq } )$ and $\mathsf { K L } ( \mathcal { S } ^ { \neq } \parallel \mathcal { S } ^ { + } ) - \mathsf { K L } ( \mathcal { S } ^ { - } \parallel \mathcal { S } ^ { + } )$ , while yielding better generalization on a range
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+ 173 of downstream tasks.
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+ 174 To ameliorate the aforementioned optimization challenge, we take a hierarchical approach. In
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+ 175 particular, we first uniformly sample a long-form content, $n \sim \mathbb { U } ( 1 , N )$ , and then draw $k$ distinct
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+ 176 snippets from ${ \mathcal { X } } _ { n }$ , creating $\{ { \bar { x } } _ { n , m } | { \bar { m } } \in { \mathcal { M } } _ { n } \}$ , where $\mathcal { M } _ { n } \subset [ 1 \cdots M _ { n } ]$ and $| { \mathcal { M } } _ { n } | = k$ . This ensures
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+ 177 that for $x _ { i } \in B$ , ${ \mathcal { N } } _ { i }$ always includes $2 k - 2$ pairs sampled from the same movie or TV show to
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+ 178 which $x _ { i }$ belongs. By putting constraints on $\mathcal { M } _ { n }$ , specifically how temporally far from each other
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+ 179 the $k$ samples are drawn, we may go one step further and to some extent control the audiovisual
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+ 180 similarity between snippets. This serves as an additional nob to tune for hard negative sampling.
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+ 181 The intuition is that, the larger narrative of a professionally made movie or TV show is composed of
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+ 182 shorter units called scene. Each scene comprises a complete event, action, or block of storytelling and
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+ 183 normally takes place in one location and deals with one action. That is, if our samples are temporally
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+ 184 close, it is more likely for corresponding snippets to be highly correlated and/or look/sound alike.
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+ 185 $k \leq \mathrm { m a x } [ \mathcal { M } _ { n } ] - \mathrm { m i n } [ \mathcal { M } _ { n } ] + 1 \ \bar { \leq } \ w \leq \bar { M } _ { n }$ defines the bounds on our sampling policy, where $w$
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+ 186 standing for a sampling window, determines the farthest two out of $k$ samples drawn from ${ \mathcal { X } } _ { n }$ can
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+ 187 be. Accordingly, $w = k$ represents the case where all $k$ samples are temporally adjacent, hence
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+ 188 the expected audiovisual similarity is maximized due to temporal continuity in content. We show
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+ 189 that having such level of hard negatives, even with a small $k$ , prevents proper training and results in
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+ 190 performance degradation. On the other hand, $w = M _ { n }$ indicates random sampling where no temporal
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+ 191 constraint is imposed on $\mathcal { M } _ { n }$ , thus samples are less likely to be drawn from adjacent time-stamps.
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+ 192 In this case, expected audiovisual similarity (i.e. hardness of negative pairs) is mainly derived from
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+ 193 global content-exclusive artifacts like, color palette, frequent appearance of the main character’s face
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+ 194 and voice, repetitive scenes, and etc. The rest of the spectrum provides middle grounds where two
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+ 195 samples drawn from ${ \mathcal { X } } _ { n }$ can at most be $w + 1$ snippets apart, something reminiscent of temporal
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+ 196 locality. Our sampling policy can be easily implemented in a few lines of Python. Please refer to
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+ 197 supplemental material for further details.
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+
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+ # 98 4 Experiments
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+
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+ # 4.1 Experimental Setup
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+
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+ Datasets and Reproducibility. We use full-length movies and episodes of TV shows for selfsupervised pretraining. Titles are randomly chosen from a large collection spanning over a variety of genres, namely Drama, Comedy, Action, Horror, Thriller, Sci-Fi and Romance. All audio is in English language. Our Movie dataset, consists of 3.6K films with an average duration of 105 minutes. Our TV dataset includes 9.2K episodes from a total of 581 shows with an average duration of 42 minutes per episode. Each of our datasets comprises 0.7 years worth of uncurated audiovisual content, which is significantly smaller than IG-Random [3] with variants at 5 and 21 years. Scaling up our pretraining datasets to volumes comparable to the IG-Random [3] while possible is non-trivial and demands dramatically larger compute resources for training, something which we currently cannot afford. Given that we cannot publicly release our dataset due to copyright reasons, we acknowledge that it is not possible for other research groups to fully reproduce our results. However, we intend to make available the pretrained models and hope that research community finds them, along with the other contributions of this work, of value whether within the context of self-supervised learning or adoption for various downstream tasks. We would like to emphasize that similar limitations have precedents in multiple earlier works including but not limited to [3, 12, 26, 43]. To evaluate the efficacy of self-supervised audio-visual representation learning from movies and TV shows, we follow recent works [3, 39, 31, 2] and benchmark UCF101[42] and HMDB51[23] for action recognition, along with ESC50[40] for audio classification. Results for the ablation studies are reported on the split-1 of the corresponding datasets. Following the standard protocol, we report the average performance over all splits when we are comparing with the state-of-the-art.
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+
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+ Pretraining. Unless mentioned otherwise, we use video snippets with 16 frames at 5 fps. For data augmentation, we resize the shorter side to 190 pixels, then randomly crop them into $1 5 8 \times 1 5 8$ pixels. As for sound, we compute mel spectrogram from the raw audio at 48K sample rate using $9 6 ~ \mathrm { m e l }$ filters and an FFT window of 2048, while the number of samples between successive frames is set to 512. For data augmentation, we randomly drop out up to $2 5 \%$ from either temporal or frequency axis of the 2-D mel spectrogram image. Training uses a batch size of 512 and takes on average 42 hours on 8 NVIDIA A100 GPUs. The dimension of audio-video joint embedding space, $d$ , is set to 512.
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+
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+ Downstream Evaluation. For training on UCF101 [42] and HMDB51 [23], we use video clips that are 32 frames long at 10 fps. Unless mentioned otherwise, these clips are randomly chosen from the duration of the video instances. A scale jittering range of [181, 226] pixels is used and we randomly crop the video into $1 5 8 \times 1 5 8$ pixels. Furthermore, random horizontal flipping and color jittering are employed. During inference, 10 temporal clips are uniformly sampled where each is spatially cropped in 3 ways (left, center, right) resulting in a total of 30 views. We then average the model predictions across these 30 views and report top-1 classification accuracy. For training on ESC50 [40], we use 3-seconds clips which are randomly chosen from the duration of the audio instances and apply time and frequency masking to spectrogram images for data augmentation. The maximum possible length of the mask is $50 \%$ of the corresponding axis. We do not use any scale jittering or random cropping on the spectrograms. During inference, 10 temporal clips are uniformly sampled
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+
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+ 238 and we average the model predictions across these 10 views and report top-1 classification accuracy.
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+ 239 For further implementation details, please refer to the supplemental material.
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+
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+ ![](images/fe3fa73a08dcc0cf4d8abc90b46bc44d9e947f10dd7a4be6b2c5ef8be9fc6fa5.jpg)
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+ Figure 1: Ablation study of the proposed sampling policy on reducing the discrepancy measures.
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+
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+ # 240 4.2 Ablation Study
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+
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+ In the following, we discuss multiple ablation studies to assess our main hypothesis that, a hierarchical sampling policy, as described in Section 3, enables better representations to be learned by increasing the portion of hard negative pairs which the contrastive loss function observes. Here, pretraining uses $90 \%$ of either Movie or TV dataset, while the remaining $10 \%$ constitute a held-out validation set7 on which we report the discrepancy measures.
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+ Sample size $( k )$ Figure 1a illustrates that compared to the baseline sampling denoted by $k = 1$ , our approach $k > 1 ,$ ) effectively shrinks the gap between $S ^ { - }$ and ${ \mathcal { S } } ^ { \neq }$ when measured either directly or against $S ^ { + }$ . Its pattern of behavior also perfectly follows our earlier intuition (ref. Section 3). In particular, given a fixed minibatch budget, a larger $k$ favors more training instances to be sampled from fewer number of long-form contents. That increases the portion of hard negative pairs, thus pushes the contrastive loss to more aggressively separate mismatched audio-video pairs from the same movie, which leads model to maintain less of the content-exclusive artifacts in the embedding space. In the most extreme case, $k = 6 4$ , all the training instances are sampled from the same movie. From Table 1, we observe that different variants of our sampling policy, with no imposed temporal constraint, i.e. $w = M _ { n }$ , outperform the baseline on all three downstream tasks
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+ ![](images/bf6588579202158118301afb6941dec79d9e5cdccd64dae9b2636f3ddcd67c90.jpg)
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+ Figure 2: Effect of color jitter on the discrepancy measures.
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+ Sampling window $\mathbf { \Pi } ^ { ( w ) }$ Smaller $w$ forces samples that belong to same movie to be drawn from a shorter temporal window, hence growing the probability that they look/sound very much alike (i.e. harder negative pairs). That is, it should further diminish the discrepancy measures. Figure 1b illustrates this behavior where we gradually increase $w$ while $k = 1 6$ . However, from Table 1, it does not seem that tuning for $w$ , i.e $w \neq M _ { n }$ , provides a meaningful gain on downstream tasks. This implies that commonalities which persist throughout the duration of a movie are sufficiently powerful signals to be exploited for generating hard negatives. We hypothesize that different scenes both within and across different movies and TV shows are of variety of length, thus a fixed $w$ is sub-optimal. Ideally, we should identify scene boundaries and and dynamically modify $w$ during sampling, something which we leave for future iterations of this work.
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+ Temporally adjacent samples. Along the lines of previous observations, Figure 1c shows that indeed drawing temporally adjacent snippets from the same long-form content, i.e. $w = k$ , results in aggressively reducing the discrepancy measures. This behavior is agnostic with respect to $k$ yet exacerbates as $k$ grows. Note that, the contrastive loss is an instance-discrimination objective function. Therefore, forcing it to distinguish between temporally adjacent snippets, that naturally sound and look extremely similar, leaves no choice for the model but to discard valuable semantic notions, which predictably leads to poor representations, also confirmed by result reported in Table 1.
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+ Table 1: Ablation study of the proposed sampling policy on different downstream tasks, measured by top-1 classification accuracy.
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+ <table><tr><td colspan="5">pretraining dataset: Movie</td></tr><tr><td>k</td><td>w</td><td>HMDB51</td><td>ESC50</td><td>UCF101</td></tr><tr><td>1</td><td>1</td><td>60.32</td><td>86.50</td><td>85.69</td></tr><tr><td>4</td><td>Mn</td><td>61.37</td><td>89.91</td><td>85.38</td></tr><tr><td>8</td><td>Mn</td><td>62.09</td><td>88.75</td><td>86.06</td></tr><tr><td>16</td><td>Mn</td><td>62.92</td><td>88.33</td><td>86.30</td></tr><tr><td>32</td><td>Mn</td><td>61.04</td><td>88.00</td><td>85.98</td></tr><tr><td>64</td><td>Mn</td><td>61.30</td><td>86.83</td><td>85.43</td></tr><tr><td>16</td><td>64</td><td>60.26</td><td>87.00</td><td>83.61</td></tr><tr><td>16</td><td>128</td><td>60.58</td><td>86.50</td><td>85.30</td></tr><tr><td>16</td><td>256</td><td>62.02</td><td>87.75</td><td>84.85</td></tr><tr><td>16</td><td>512</td><td>61.30</td><td>87.08</td><td>85.38</td></tr><tr><td>16</td><td>1024</td><td>60.65</td><td>86.16</td><td>84.61</td></tr><tr><td>16</td><td>2048</td><td>61.83</td><td>87.66</td><td>85.11</td></tr><tr><td>4</td><td>4</td><td>60.19</td><td>88.00</td><td>84.66</td></tr><tr><td>16</td><td>16</td><td>56.86</td><td>88.75</td><td>82.71</td></tr><tr><td>64</td><td>64</td><td>57.45</td><td>84.58</td><td>82.68</td></tr><tr><td colspan="5">pretraining dataset:TV</td></tr><tr><td>k</td><td>w</td><td>HMDB51</td><td>ESC50</td><td>UCF101</td></tr><tr><td>1</td><td>1</td><td>56.40</td><td>85.50</td><td>84.37</td></tr><tr><td>8</td><td>Mn</td><td>61.50</td><td>87.50</td><td>85.96</td></tr><tr><td>16</td><td>Mn</td><td>61.69</td><td>89.00</td><td>85.64</td></tr><tr><td>8</td><td>64</td><td>60.58</td><td>88.00</td><td>85.96</td></tr><tr><td>8</td><td>128</td><td>60.00</td><td>85.66</td><td>85.77</td></tr><tr><td>16</td><td>256</td><td>61.30</td><td>86.41</td><td>85.01</td></tr></table>
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+ Movies vs. TV Shows. To confirm that our sampling policy behaves consistently across both movies and TV shows, Figure 1d illustrates the discrepancy measures computed on TV dataset. We observe similar effectiveness when using $k$ and $w$ as tuning nobs for reducing either ${ \mathsf { K L } } ( S ^ { - } \parallel S ^ { \neq } )$ or the gap between ${ \mathsf { K L } } ( S ^ { - } \parallel S ^ { + } )$ and $\mathsf { K L } ( S ^ { \neq } \parallel S ^ { + } )$ . Table 1 demonstrates that different variants of our approach significantly outperform the baseline, i.e. $k = 1$ . We attribute the larger gains achieved when using TV instead of Movie dataset to the fact that content diversity is naturally lower when pretraining on TV shows since each one includes many episodes that all are characterized with the same content-exclusive artifacts.
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+ Color jitter. We have established so far that commonalities which persist throughout the duration of a long-form content, things likely associated with color pallet, frequent appearance of the main character’s face and voice, and repetitive scenes can be exploited for learning better representations. That is, one may naturally assume that employing data augmentation techniques like color jitter should be helpful since by distorting content-exclusive visual artifacts, color jitter is expected to reduce ${ \mathsf { K L } } ( S ^ { - } \parallel S ^ { \neq } )$ . Figure 2 illustrates the effect of color jitter, where brightness, contrast, and saturation jitter values are chosen uniformly from $\mathtt { [ m a x ( 0 , 1 - \sigma ) , 1 + \sigma ] }$ . We observe that color jitter reduces the discrepancy measures for the baseline but not as much as it can be obtained by our proposed sampling policy $( k > 1 )$ ), and even then according to Table 2 only yields a slight gain on downstream tasks.
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+ $\ell _ { 2 }$ -normalized feature space. The common practice [2, 31, 39, 7] is to compute contrastive loss in $\ell _ { 2 }$ -normalized feature space, where according to [47] the temperature hyper-parameter, $\tau$ , controls the strength of penalties on hard negative samples. We explored this with two widely-used $\tau$ values. From Table 2, we observe that compared to operating in an unnormalized embedding space, adopting such design choice results in a large performance drop on HMDB51[23] while other downstream benchmarks see only negligible gains.
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+ Table 4: Effect of self-supervised learning from curated versus uncurated data on different downstream tasks. The “years” column indicates the duration of the pretraining datasets in years.
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+ <table><tr><td>method</td><td>pretraining dataset</td><td>uncurated</td><td>years</td><td>HMDB51</td><td>ESC50</td><td>UCF101</td></tr><tr><td>Ours</td><td>Movie</td><td></td><td>0.7</td><td>62.9</td><td>88.3</td><td>86.3</td></tr><tr><td>Ours</td><td>TV</td><td></td><td>0.7</td><td>61.7</td><td>89.0</td><td>85.6</td></tr><tr><td>XDC[3]</td><td>IG-Random16M</td><td>√</td><td>5</td><td>55.2</td><td>84.3</td><td>84.1</td></tr><tr><td>XDC[3]</td><td>IG-Random65M</td><td>√</td><td>21</td><td>61.2</td><td>86.3</td><td>88.8</td></tr><tr><td>XDC[3]</td><td>IG-Kinetics16M</td><td>×</td><td>5</td><td>57.3</td><td>82.5</td><td>87.6</td></tr><tr><td>XDC[3]</td><td>IG-Kinetics65M</td><td>×</td><td>21</td><td>63.1</td><td>84.8</td><td>91.5</td></tr></table>
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+ Curated vs. Uncurated data. To the best of our knowledge, the only other uncurated dataset used for audio-visual self-supervised learning is IGRandom[3]8. Table 4 confirms that learning from uncurated movies and TV shows is extremely effective. Our results significantly exceed those of XDC[3] obtained on IG-Random16M despite using a simpler model and 7 times smaller volume of pretraining data. Even in comparison to IG-Random65M with 30 times larger data, we obtain better performances on 2 out of 3 benchmarks. The most promising of our findings though is how competitive our results are against XDC[3] when it is trained on variants of IG-Kinetics which are not only curated but also orders of magnitude larger. With all that, we confidently reject the notion that audio-visual self-supervised learning from uncurated data considerably lags behind utilizing large-scale curated datasets.
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+ Effect of genre. The distribution of genre among movies used in our pretraining is the closest we have to taxonomy in the curated datasets. So, it is worth examining the quality of our learned representations under various genre distributions. To do so, given a fixed pretraining budget $( N = 1 . 6 \mathsf { K } )$ , we compare four different scenarios where movies used in the pretraining are distributed i) non-uniformly over all genres except Drama, and Comedies, ii) non-uniformly over Drama, and Comedies, iii) uniformly over all genres, and iv) non-uniformly over all genres. Table 3 confirms that indeed there is very little difference between the aforementioned setups when it comes to transfer learning to the downstream tasks.
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+ Table 2: Effect of color jitter $( \sigma )$ and computing contrastive loss in $\ell _ { 2 }$ -normalized embedding space with temperature hyper-parameter $( \tau )$ on different downstream tasks.
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+ <table><tr><td>k</td><td>0</td><td>HMDB51</td><td>ESC50</td><td>UCF101</td></tr><tr><td>1</td><td>0.0</td><td>60.32</td><td>86.50</td><td>85.69</td></tr><tr><td>16</td><td>0.0</td><td>62.92</td><td>88.33</td><td>86.30</td></tr><tr><td>1</td><td>1.0</td><td>60.45</td><td>87.66</td><td>84.82</td></tr><tr><td>16</td><td>0.5</td><td>60.13</td><td>87.75</td><td>85.98</td></tr><tr><td>16</td><td>1.0</td><td>61.11</td><td>88.33</td><td>85.93</td></tr><tr><td>k</td><td>T</td><td>HMDB51</td><td>ESC50</td><td>UCF101</td></tr><tr><td>16</td><td>0.07</td><td>60.78</td><td>87.08</td><td>86.86</td></tr><tr><td>16</td><td>0.30</td><td>60.78</td><td>89.25</td><td>85.72</td></tr></table>
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+ Table 3: Effect of genre distribution in Movie dataset on different downstream tasks. Experiments are conducted with input spatial resolution of $1 1 2 \times 1 1 2$ pixels.
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+ <table><tr><td>setting</td><td>HMDB51</td><td>ESC50</td><td>UCF101</td></tr><tr><td>i</td><td>57.58</td><td>86.50</td><td>82.44</td></tr><tr><td>ii</td><td>56.99</td><td>85.50</td><td>82.39</td></tr><tr><td>i</td><td>56.27</td><td>85.25</td><td>82.87</td></tr><tr><td>iv</td><td>56.40</td><td>86.75</td><td>83.24</td></tr></table>
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+ # 4.3 Comparison with state-of-the-art
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+ Table 5 compares our proposed approach of learning from Movies and TV shows against the best performing audio-visual self-supervised learning methods. In general, our numbers are comparable with the best existing results reported in the literature, even with much less data and considerably simpler model/training procedure9. It is interesting that training on Movie dataset alone obtains
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+ Table 5: Comparison with state-of-the-art. Dataset abbreviations: AudioSet[11], HowTo100M[28], IG-Kinetics65M [12]; their length in years is given in the “years” column. “Arch.” denotes the architecture of video backbone $( f )$ . [2]† indicates when the corresponding model use only audio and video, and not text modality. For a fair comparison, when using only Movie dataset, we train for twice as many epochs as our other variants in order to match their total number of gradient updates.
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+ <table><tr><td>Method</td><td>Arch.</td><td>pretraining dataset</td><td>curated</td><td>years</td><td>HMDB51</td><td>UCF101</td><td>ESC50</td></tr><tr><td>GDT[39]</td><td>R(2+1)D-18</td><td>AS</td><td></td><td>1</td><td>66.1</td><td>92.5</td><td>88.5</td></tr><tr><td>GDT[39]</td><td>R(2+1)D-18</td><td>IG65M</td><td></td><td>21</td><td>72.8</td><td>95.2</td><td></td></tr><tr><td>XDC[3]</td><td>R(2+1)D-18</td><td>AS</td><td>√</td><td>1</td><td>61.0</td><td>91.2</td><td>84.8</td></tr><tr><td>XDC[3]</td><td>R(2+1)D-18</td><td>IG65M</td><td>√</td><td>21</td><td>67.4</td><td>94.2</td><td></td></tr><tr><td>AVTS[22]</td><td>MC3</td><td>AS</td><td></td><td>1</td><td>61.6</td><td>89.0</td><td>82.3</td></tr><tr><td>AVID[31]</td><td>R(2+1)D-18</td><td>AS</td><td></td><td>1</td><td>64.7</td><td>91.5</td><td>89.1</td></tr><tr><td>MMV[2]+</td><td>R(2+1)D-18</td><td>AS</td><td>√</td><td>1</td><td>70.1</td><td>91.5</td><td>85.6</td></tr><tr><td>MMV[2]+</td><td>S3D-G</td><td>AS</td><td>√</td><td>1</td><td>68.2</td><td>90.1</td><td>86.1</td></tr><tr><td>MMV[2]+</td><td>S3D-G</td><td>AS+HT</td><td>√</td><td>16</td><td>68.3</td><td>91.1</td><td>87.2</td></tr><tr><td>Ours (k=16)</td><td>R(2+1)D-18</td><td>Movie</td><td>X</td><td>0.7</td><td>64.5</td><td>87.9</td><td>88.8</td></tr><tr><td>Ours (k=8)</td><td>R(2+1)D-18</td><td>Movie+TV</td><td>X</td><td>1.4</td><td>65.0</td><td>87.7</td><td>89.1</td></tr><tr><td>Ours (k=16)</td><td>R(2+1)D-18</td><td>Movie+TV</td><td>X</td><td>1.4</td><td>65.1</td><td>88.5</td><td>89.1</td></tr><tr><td>Ours (k=32)</td><td>R(2+1)D-18</td><td>Movie+TV</td><td>X</td><td>1.4</td><td>65.6</td><td>88.7</td><td>88.2</td></tr></table>
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+ 358 comparable performance to the cases where both TV and Movie datasets are used for pretraining.
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+ 359 This further confirms the richness of the training data which movies and TV shows can provide
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+ 360 to self-supervised learning problems. We also see that increasing $k$ even beyond 8 gives further
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+ 361 incremental gains on action recognition benchmarks.
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+ # 362 5 Conclusion
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+ Despite its amazing recent progress, state-of-the-art self-supervised learning still heavily relies on supervised, i.e. curated, large-scale datasets for pretraining. In this work, we have shown that pretraining solely on uncurated data in forms of movies and TV shows, even at a comparatively small scale, can give rise to representations which are capable of competing with the state-of-theart of more complex architectures trained on larger curated datasets. This comes contrary to the current literature which tends to suggest that learning from uncurated data largely falls behind the use of curated alternatives. We intentionally made design decisions to keep our approach and training strategy as simple as possible to demonstrate that learning decently powerful audio-visual representations does not necessarily require gigantic data and compute resources. Through extensive set of experiments, our work establishes for the first time the efficacy of self-supervised learning of audio-visual representations from movies and TV shows.
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+ # 374 6 Broader impact
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+ Potential benefits. Our work shows that competitive multimodal representations can be learned from a comparatively small volume of uncurated data in the form of movies and TV shows. Besides minimizing any sort of human-involvement, which we believe must have already been paid an extra attention to in the literature, our work demonstrates that one does not require gigantic data and compute resources for effective self-supervised pretraining. Such results promise a more democratized research arena where smaller groups are not alienated due lack of sufficient compute resources. More importantly, lowering the compute requirements naturally reduces any environmental effects which training these models can potentially have.
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+ Potential risks. Any machine learning method is susceptible to the potential underlying biases in the data. This is more important for self-supervised methods that deal with huge volumes, often not evaluated by diverse group of humans for any fairness concerns. The same is generally true in our case which requires us to make sure that titles that are included in training are diverse and inclusive.
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+ 387 References
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+ # Checklist
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+ (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]
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+ (b) Did you describe the limitations of your work? [Yes] Specifically, training data being proprietary creates concerns around reproducibility, which has precedence in the literature as mentioned in the paper. We address that partially by planning to publicly release pretrained models.
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+ (c) Did you discuss any potential negative societal impacts of your work? [Yes]
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+ (d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
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+ 2. If you are including theoretical results...
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+ (a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
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+ 3. If you ran experiments...
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+ (a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No] The data is proprietary. However, we have provided implementation of the proposed method in supplemental material and aim to publicly release the pretrained models.
398
+ (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] They are all discussed in detail either in the main submission or in supplemental material.
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+ (c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We only observed meaningful differences after running experiments multiple times, for ESC50[40] downstream experiments. Corresponding standard errors are reported in supplemental material.
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+ (d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] Please refer to Section 4.1.
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+ (a) If your work uses existing assets, did you cite the creators? [N/A]
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+ (b) Did you mention the license of the assets? [N/A]
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+ (c) Did you include any new assets either in the supplemental material or as a URL? [N/A]
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+ (d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]
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+ (e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
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+ 5. If you used crowdsourcing or conducted research with human subjects...
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+ (a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
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+ (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
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+ (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]
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+ "text": "1 The abundance and ease of utilizing sound, along with the fact that auditory clues \n2 reveal so much about what happens in the scene, make the audio-visual space a \n3 perfectly intuitive choice for self-supervised representation learning. However, \n4 the current literature suggests that training on uncurated data yields considerably \n5 poorer representations compared to the curated alternatives collected in supervised \n6 manner, and the gap only narrows when the volume of data significantly increases. \n7 Furthermore, the quality of learned representations is known to be heavily influ \n8 enced by the size and taxonomy of the curated datasets used for self-supervised \n9 training. This begs the question of whether we are celebrating too early on catching \n10 up with supervised learning when our self-supervised efforts still rely almost exclu \n11 sively on curated data. In this paper, we study the efficacy of learning from Movies \n12 and TV Shows as forms of uncurated data for audio-visual self-supervised learning. \n13 We demonstrate that a simple model based on contrastive learning, trained on a \n14 collection of movies and TV shows, not only dramatically outperforms more com \n15 plex methods which are trained on orders of magnitudes larger uncurated datasets, \n16 but also performs very competitively with the state-of-the-art that learns from \n17 large-scale curated data. We identify that audiovisual patterns like the appearance \n18 of the main character or prominent scenes and mise-en-scène which frequently \n19 occur through the whole duration of a movie, lead to an overabundance of easy \n20 negative instances in the contrastive learning formulation. Capitalizing on such \n21 observation, we propose a hierarchical sampling policy, which despite its simplicity, \n22 effectively improves the performance, particularly when learning from TV shows \n23 which naturally face less semantic diversity. ",
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+ "text": "25 Recently, there has been tremendous progress in self-supervised learning from still images, where the \n26 standard supervised training has been outperformed in a variety of image-related tasks [7, 8, 15, 29]. \n27 The appeal of detaching representation learning from human annotations is rooted not only in the \n28 non-trivial challenges of scaling-up the labeling process, but also in the ill-defined task of determining \n29 a proper taxonomy with generalization power and transferability. Both challenges only exacerbate as \n30 we move from images to videos, where the notion of time is involved and the complexity of visual \n31 concepts increases. Simply considering the number of training instances or even the cardinality of \n32 the label set is not sufficient to conclude if one large-scale supervised dataset is more suitable than \n33 another for transfer learning in video classification tasks [20]. That is, the abundance of attention \n34 which video self-supervised learning has lately received is only to be expected. While many research \n35 efforts in this area extend the contributions made initially in the image domain to the video domain, \n36 others, including our work, have explored harnessing additional modalities such as audio or text for \n37 multi-modal self-supervised learning [2, 3, 4, 22, 27, 31, 37, 36, 39]. \n38 From the current state-of-the-art one makes two major conclusions. First, the quality of learned \n39 representations, evaluated by fine-tuning on downstream tasks, is heavily influenced by the size and \n40 taxonomy of the pretraining datasets [2, 3, 39]. Second, an uncurated pretraining dataset yields \n41 considerably poorer representations compared to a curated one and the gap only narrows when the \n42 total amount of pretraining data significantly increases [3]. Curated data refers to likes of supervised \n43 large-scale action recognition and audio classification datasets such as Kinetics [6], IG-Kinetics \n44 [12], AudioSet [11], and YouTube-8M [1]. While the human-annotated labels are not accessed for \n45 self-supervised pretraining, videos being trimmed and from a label set of limited cardinality with \n46 biased sampling distribution1 implicitly acts as a sort of supervision. On the other hand, an uncurated \n47 data refers to likes of IG-Random[3], simply a body of unlabeled videos collected blindly with \n48 none of the aforementioned careful human-involvements. That being said, we know that something \n49 as simple as having access to a clean object-centric training data, like Imagenet, can be indirectly \n50 exploited by contrastive self-supervised learning in image domain to obtain additional performance \n51 gain [41] on the downstream tasks which exhibit similar properties. The analogous to it of course are \n52 the well trimmed closed-set curated datasets which are being extensively used in the literature for \n53 video self-supervised pretraining, while downstream evaluations focus on benchmarks with similar \n54 characteristics. Our work aims at comprehensively exploring the efficacy of learning from Movies \n55 and TV Shows, as forms of uncurated data, for audio-visual self-supervised learning. \n56 Many of us can relate to an experience in movie theaters when the sound of the engine, first perceived \n57 by our left ear, is gradually heard more by the right ear as a car moves from the left side of the \n58 screen to the right side. Another example is a scene in which an object, like a helicopter, approaches \n59 the camera from distance and eventually flies over it. In this case, the perceived sound not only \n60 changes in loudness but also transitions from front to back, in concert with the visuals, giving the \n61 audience a more realistic feeling as if they are indeed positioned behind the camera. Besides, with \n62 art being inherently novel, two movies even if they share genres or revolve around similar story \n63 lines often deliver quite different experiences and portray distinct visuals, thanks to the extremely \n64 artist-driven creative process behind such productions. We hypothesize that the aforementioned high \n65 audio fidelity, and inherent semantic diversity characterize long-form content2 as potentially a very \n66 rich source for self-supervised multi-modal representation learning. It is worth emphasizing that in \n67 spirit of uncurated data, we not only blindly sample from a large collection of movies and TV shows \n68 when constructing our pretraining dataset, but also perform ablation studies on the effect of genre \n69 distribution, the closest we have to taxonomy in the curated datasets, confirming that the quality of \n70 learned representations is agnostic with respect to such statistics. \n71 To the best of our knowledge, we are the first to solely rely on uncurated data and study the efficacy of \n72 self-supervised multi-modal representation learning from movies and TV shows. Despite meaningful \n73 domain gap between our pretraining data and the space of downstream tasks, we obtain representations \n74 which are very competitive with those learned from curated datasets. This is particularly important as \n75 we follow a much simpler modeling approach in comparison with the state-of-the art. ",
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+ "text": "77 Self-supervised learning techniques define pretext tasks, mostly inspired by the natural structures \n78 in the data, in order to generate supervisory signals for training. Despite the plethora of proposed \n79 pretext tasks in the literature, these approaches can be coarsely divided into two groups, namely \n80 pretext learning, and pretext-invariant methods. Approaches which fall in the former bucket, usually \n81 apply a form of transform, randomly drawn from a parametric family, to the input data then optimize \n82 for predicting the parameters of the chosen transformation. Predicting the relative position of image \n83 patches [9], solving jigsaw puzzles [33], estimating artificial rotations [13], colorization [50], context \n84 encoders learned through inpainting [38], and learning by counting scale and split invariant visual \n85 primitives [34], are among many methods which belong to this category. Similar techniques have \n86 been extended from images to videos [10, 21, 24, 25, 30, 46, 48, 49], where in addition to the \n87 spatial context, the temporal domain, and the arrow of time have been heavily exploited. In contrast, \n88 pretext-invariant methods [5, 7, 8, 15, 18, 17, 29, 35, 39, 44] are built on the concept of maximizing \n89 mutual information across augmented versions of a single instance, and are mostly formulated as \n90 contrastive learning. In other words, a pretext is used to generate different views of a single input \n91 for which the learning algorithm aims to maximize the intra-instance similarity, across variety of \n92 transformations. Our work falls within this category, however we function in a multi-modal realm \n93 employing both audio and video. \n94 Earlier works which harnessed audio and video for representation learning, have leveraged audio \n95 visual temporal synchronization [22, 36], correspondence [4], and cross-modal clustering [3, 37]. The \n96 work by Patrick et al.[39] proposes a generalized data transformation in order to unify a variety of \n97 audio-visual self-supervised pretext tasks through a noise contrastive formulation. This work is close \n98 to ours in choice of objective function and data type, yet we employ no augmentation (except modality \n99 projection in the terminology of [39]), and solely focus on capitalizing the advantages of learning \n100 from long-form content. Morgado et al.[31] show that cross-modal discrimination is important for \n101 learning good audio and video representations, something which was also pointed out earlier in a \n102 clustering framework [3]. Beyond that, [31] generalizes the notion of instance-level positive and \n103 negative examples by exploring cross-modal agreement where multiple instances are grouped together \n104 as positives by measuring their similarity in both the video and audio feature spaces. While we also \n105 adopt a cross-modal noise contrastive estimation loss, we stick with the vanilla version, instance-level \n106 positive and negatives, and do not use any memory bank feature representations. Finally, Alayrac et \n107 al.[2] recently proposed a multi-modal versatile network capable of simultaneously learning from \n108 audio, video and text. Building on the intuition that different modalities are of different semantic \n109 granularity, audio and video are first compared in a fine-grained space while text is compared with \n110 the aforementioned modalities in a lower dimensional coarse-grained space. In our experiments, we \n111 compare with a variant of [2] where only audio and video modalities are utilized. ",
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+ "text": "113 Notations and Architecture. Our pretraining dataset is denoted by $\\mathcal { X } \\ : = \\ : \\{ \\mathcal { X } _ { n } | n \\in [ 1 \\cdot \\cdot \\cdot N ] \\}$ \n114 where $\\mathcal { X } _ { n } = \\{ x _ { n , m } | m \\in [ 1 \\cdots M _ { n } ] \\}$ contains $M _ { n }$ non-overlapping audiovisual snippets which are \n115 temporally segmented from the duration of the $n ^ { t h }$ long-form content in the dataset. Each snippet \n116 includes both audio and video modalities, formally $x _ { n , m } = ( a _ { n , m } , v _ { n , m } )$ , where $a _ { n , m } \\in \\mathbb { R } ^ { 1 \\times \\pmb { \\tilde { P } } \\times Q }$ \n117 and $v _ { n , m } \\in \\mathbb { R } ^ { 3 \\times T \\times H \\times W }$ . $T , H$ , and $W$ denote the number of frames, height and width of the video, \n118 while $P$ , and $Q$ respectively stand for the number of mel filters, and audio frames. Video and audio are \n119 processed through 18-layers deep $\\mathrm { R } ( 2 { + } 1 ) \\mathrm { D }$ [45] and ResNet [16] architectures, respectively referred \n120 to as $f : \\mathbb { R } ^ { 3 } \\check { \\mathbb { R } } ^ { d _ { f } }$ and $\\dot { \\boldsymbol { g } } : \\mathbb { R } ^ { 1 } \\dot { } \\mathbb { R } ^ { d _ { g } }$ . Inspired by [7], we use projection heads, $\\boldsymbol { h } _ { f } : \\mathbb { R } ^ { d _ { f } } \\overset { \\cdot } { } \\mathbb { R } ^ { d }$ and \n121 $h _ { g } : \\mathbb { R } ^ { d _ { g } } \\mathbb { R } ^ { d }$ , to map corresponding representations into a common $d$ -dimensional space before \n122 computing the contrastive loss. The shallow architecture of $h _ { f }$ and $h _ { g }$ consists of two convolution \n123 layers, separated by Batch Normalization [19] and ReLU [32], followed by global average pooling. \n124 Once self-supervised pretraining finished, we discard the projection heads and fine-tune $f$ and $g$ for \n125 respective downstream tasks. \n126 Loss Function. With a slight abuse of notation3, $\\boldsymbol { B } = \\{ x _ { i } = ( a _ { i } , v _ { i } ) | i \\in [ 1 \\cdot \\cdot \\cdot B ] \\}$ represents a \n127 minibatch of size $B$ , where video and audio modalities associated with the $i ^ { t h }$ sample, $x _ { i }$ , are denoted \n128 by $v _ { i }$ and $a _ { i }$ . We use $z _ { v } ^ { i } = h _ { f } ( f ( v _ { i } ) )$ and $z _ { a } ^ { i } = h _ { g } ( g ( a _ { i } ) )$ to represent the associated embeddings \n129 generated by projection heads, and optimize the noise-contrastive loss [14] shown in 1 in order \n130 to maximize the symmetric joint probability between audio and video. For the $i ^ { t h }$ element in the \n131 minibatch, $( z _ { v } ^ { i } , z _ { a } ^ { i } )$ serves as the positive pair, while assuming negative pairs for both modalities, \n132 $\\mathcal { N } _ { i } = \\{ ( z _ { v } ^ { i } , z _ { a } ^ { j } ) , ( \\stackrel { \\sim } { z } _ { v } ^ { j } , z _ { a } ^ { i } ) | j \\in [ 1 \\cdots B ] , i \\neq j \\}$ constitutes the set of negative pairs. ",
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+ "text": "133 Most of the previous works [2, 31, 39] normalize the embeddings before computing the contrastive \n134 loss and employ a temperature hyper-parameter, often denoted by $\\tau$ as in [2, 31], to control the \n35 smoothness for the distribution of pairwise similarities. In contrast, we have chosen to operate in an \n36 unnormalized embedding space. Besides the obvious benefit of eliminating the need for tuning $\\tau$ , we \n7 empirically show that such decision does not affect the quality of the learned representations. \n138 Sampling Policy. Contrastive loss function shown in Equation 1 is computed over $B$ training \n139 instances, each in form of an audiovisual snippet. A naive sampling policy may ignore the fact that \n140 snippets comprising the pretraining dataset are in fact temporal segments that were trimmed from \n141 longer-form contents, i.e. movies and TV shows. Such an assumption treats our training data as \n142 independent and identically distributed random variables from $\\textstyle \\bigcup _ { n = 1 } ^ { N } { \\mathcal { X } } _ { n }$ , which constitutes the default \n143 sampling policy that is commonly used in the general deep learning literature. However, in reality, \n144 commonalities and correlations do exist along the temporal axis of a movie or TV show, things like \n145 audio mastering artifacts, frequent appearance of the main character’s face and voice, thematic music, \n146 repetitive scenes and mise-en-scène4, all of which contribute to breaking the previously discussed \n147 i.i.d assumption. This is even more pronounced when we deal with multiple episodes of the same TV \n148 show appearing in the pretraining dataset5. Note that, sampling from no video data is going to be i.i.d \n149 but in this case the temporal correlations extend for much longer given our entities are movies and TV \n150 shows. Thus, it is more accurate to think of $\\mathcal { X }$ having multiple underlying domains, oriented towards \n151 exclusive properties which different long-form contents are characterized by. We hypothesize that \n152 during training, model gradually discovers such patterns of commonalities, which are not semantically \n153 valuable, and latches onto those to quickly minimize Equation 1 leading to poor generalization6. The \n154 reason being $B \\ll N$ , hence for $n \\sim \\mathbb { U } ( 1 , N )$ and $m \\neq m ^ { \\prime }$ , $\\mathsf { P } ( x _ { n , m } \\in B \\land x _ { n , m ^ { \\prime } } \\in B )$ is negligible. \n155 In other words, the set of negative pairs in Equation 1 mainly includes pairs for which audio and \n156 video come from two different movies or TV shows, thus due to the aforementioned artifacts behave \n157 as easy negatives. \n158 In order to quantitatively measure our hypothesis, we define different distributions, shown in Equation \n159 2, over the space of audio-visual similarity. $S ^ { + }$ indicates the space of correct matches, i.e. where \n160 audio and video correspond to the same snippet. $S ^ { - }$ indicates the space where audio and video do \n161 not correspond yet belong to the same movie or TV show. Finally, $\\bar { \\boldsymbol { S } } ^ { \\neq }$ indicates the space in which \n162 audio and video are sampled from two distinct long-form content, hence naturally do not correspond. ",
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+ "text": "$$\n( z _ { v } ^ { n , m } ) ^ { \\mathsf { T } } ( z _ { a } ^ { n ^ { \\prime } , m ^ { \\prime } } ) \\sim \\left\\{ { \\begin{array} { l l } { S ^ { + } , } & { { \\mathrm { i f } } \\ n = n ^ { \\prime } \\wedge m = m ^ { \\prime } } \\\\ { S ^ { - } , } & { { \\mathrm { i f } } \\ n = n ^ { \\prime } \\wedge m \\neq m ^ { \\prime } } \\\\ { S ^ { \\neq } , } & { { \\mathrm { i f } } \\ n \\neq n ^ { \\prime } \\wedge \\forall ( m , m ^ { \\prime } ) } \\end{array} } \\right.\n$$",
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+ "text": "163 With that, and $\\mathsf { K L }$ denoting Kullback–Leibler divergence, $\\mathsf { K L } ( S ^ { - } \\parallel S ^ { + } )$ measures the expected \n164 difference between positive and negative pairs within the same movie or TV show. Ideally, this should \n165 increase as the training progresses, since the model gradually learns audio-video correspondence by \n166 minimizing Equation 1. Meanwhile, the i.i.d assumption suggests $\\mathsf { K L } ( \\mathcal { S } ^ { - } \\parallel \\mathcal { S } ^ { + } ) \\simeq \\mathsf { K L } ( \\mathcal { S } ^ { \\neq } \\parallel \\mathcal { S } ^ { + } )$ \n167 and $\\mathsf { K L } ( \\mathcal { S } ^ { - } \\parallel \\mathcal { S } ^ { \\neq } ) \\simeq \\boldsymbol { 0 }$ , yet as we empirically illustrate later, $\\mathsf { K L } ( \\mathcal { S } ^ { - } \\parallel \\mathcal { S } ^ { + } ) < \\mathsf { K L } ( \\mathcal { S } ^ { \\neq } \\parallel \\mathcal { S } ^ { + } )$ and \n168 ${ \\mathsf { K L } } ( S ^ { - } \\parallel S ^ { \\neq } )$ is rather large, indicating that, upon convergence and on a held-out set, model has \n169 a harder time pushing apart negative pairs when audio and video come from the same underlying \n170 long-form content. Next, we explain how a simple alternative policy which samples $k$ snippets \n171 from each long-form content effectively reduces both of the discrepancy measures, referring to \n172 $\\mathsf { K L } ( S ^ { - } \\parallel S ^ { \\neq } )$ and $\\mathsf { K L } ( \\mathcal { S } ^ { \\neq } \\parallel \\mathcal { S } ^ { + } ) - \\mathsf { K L } ( \\mathcal { S } ^ { - } \\parallel \\mathcal { S } ^ { + } )$ , while yielding better generalization on a range \n173 of downstream tasks. \n174 To ameliorate the aforementioned optimization challenge, we take a hierarchical approach. In \n175 particular, we first uniformly sample a long-form content, $n \\sim \\mathbb { U } ( 1 , N )$ , and then draw $k$ distinct \n176 snippets from ${ \\mathcal { X } } _ { n }$ , creating $\\{ { \\bar { x } } _ { n , m } | { \\bar { m } } \\in { \\mathcal { M } } _ { n } \\}$ , where $\\mathcal { M } _ { n } \\subset [ 1 \\cdots M _ { n } ]$ and $| { \\mathcal { M } } _ { n } | = k$ . This ensures \n177 that for $x _ { i } \\in B$ , ${ \\mathcal { N } } _ { i }$ always includes $2 k - 2$ pairs sampled from the same movie or TV show to \n178 which $x _ { i }$ belongs. By putting constraints on $\\mathcal { M } _ { n }$ , specifically how temporally far from each other \n179 the $k$ samples are drawn, we may go one step further and to some extent control the audiovisual \n180 similarity between snippets. This serves as an additional nob to tune for hard negative sampling. \n181 The intuition is that, the larger narrative of a professionally made movie or TV show is composed of \n182 shorter units called scene. Each scene comprises a complete event, action, or block of storytelling and \n183 normally takes place in one location and deals with one action. That is, if our samples are temporally \n184 close, it is more likely for corresponding snippets to be highly correlated and/or look/sound alike. \n185 $k \\leq \\mathrm { m a x } [ \\mathcal { M } _ { n } ] - \\mathrm { m i n } [ \\mathcal { M } _ { n } ] + 1 \\ \\bar { \\leq } \\ w \\leq \\bar { M } _ { n }$ defines the bounds on our sampling policy, where $w$ \n186 standing for a sampling window, determines the farthest two out of $k$ samples drawn from ${ \\mathcal { X } } _ { n }$ can \n187 be. Accordingly, $w = k$ represents the case where all $k$ samples are temporally adjacent, hence \n188 the expected audiovisual similarity is maximized due to temporal continuity in content. We show \n189 that having such level of hard negatives, even with a small $k$ , prevents proper training and results in \n190 performance degradation. On the other hand, $w = M _ { n }$ indicates random sampling where no temporal \n191 constraint is imposed on $\\mathcal { M } _ { n }$ , thus samples are less likely to be drawn from adjacent time-stamps. \n192 In this case, expected audiovisual similarity (i.e. hardness of negative pairs) is mainly derived from \n193 global content-exclusive artifacts like, color palette, frequent appearance of the main character’s face \n194 and voice, repetitive scenes, and etc. The rest of the spectrum provides middle grounds where two \n195 samples drawn from ${ \\mathcal { X } } _ { n }$ can at most be $w + 1$ snippets apart, something reminiscent of temporal \n196 locality. Our sampling policy can be easily implemented in a few lines of Python. Please refer to \n197 supplemental material for further details. ",
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+ "text": "98 4 Experiments ",
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+ "text": "Datasets and Reproducibility. We use full-length movies and episodes of TV shows for selfsupervised pretraining. Titles are randomly chosen from a large collection spanning over a variety of genres, namely Drama, Comedy, Action, Horror, Thriller, Sci-Fi and Romance. All audio is in English language. Our Movie dataset, consists of 3.6K films with an average duration of 105 minutes. Our TV dataset includes 9.2K episodes from a total of 581 shows with an average duration of 42 minutes per episode. Each of our datasets comprises 0.7 years worth of uncurated audiovisual content, which is significantly smaller than IG-Random [3] with variants at 5 and 21 years. Scaling up our pretraining datasets to volumes comparable to the IG-Random [3] while possible is non-trivial and demands dramatically larger compute resources for training, something which we currently cannot afford. Given that we cannot publicly release our dataset due to copyright reasons, we acknowledge that it is not possible for other research groups to fully reproduce our results. However, we intend to make available the pretrained models and hope that research community finds them, along with the other contributions of this work, of value whether within the context of self-supervised learning or adoption for various downstream tasks. We would like to emphasize that similar limitations have precedents in multiple earlier works including but not limited to [3, 12, 26, 43]. To evaluate the efficacy of self-supervised audio-visual representation learning from movies and TV shows, we follow recent works [3, 39, 31, 2] and benchmark UCF101[42] and HMDB51[23] for action recognition, along with ESC50[40] for audio classification. Results for the ablation studies are reported on the split-1 of the corresponding datasets. Following the standard protocol, we report the average performance over all splits when we are comparing with the state-of-the-art. ",
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+ "text": "Pretraining. Unless mentioned otherwise, we use video snippets with 16 frames at 5 fps. For data augmentation, we resize the shorter side to 190 pixels, then randomly crop them into $1 5 8 \\times 1 5 8$ pixels. As for sound, we compute mel spectrogram from the raw audio at 48K sample rate using $9 6 ~ \\mathrm { m e l }$ filters and an FFT window of 2048, while the number of samples between successive frames is set to 512. For data augmentation, we randomly drop out up to $2 5 \\%$ from either temporal or frequency axis of the 2-D mel spectrogram image. Training uses a batch size of 512 and takes on average 42 hours on 8 NVIDIA A100 GPUs. The dimension of audio-video joint embedding space, $d$ , is set to 512. ",
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+ "text": "Downstream Evaluation. For training on UCF101 [42] and HMDB51 [23], we use video clips that are 32 frames long at 10 fps. Unless mentioned otherwise, these clips are randomly chosen from the duration of the video instances. A scale jittering range of [181, 226] pixels is used and we randomly crop the video into $1 5 8 \\times 1 5 8$ pixels. Furthermore, random horizontal flipping and color jittering are employed. During inference, 10 temporal clips are uniformly sampled where each is spatially cropped in 3 ways (left, center, right) resulting in a total of 30 views. We then average the model predictions across these 30 views and report top-1 classification accuracy. For training on ESC50 [40], we use 3-seconds clips which are randomly chosen from the duration of the audio instances and apply time and frequency masking to spectrogram images for data augmentation. The maximum possible length of the mask is $50 \\%$ of the corresponding axis. We do not use any scale jittering or random cropping on the spectrograms. During inference, 10 temporal clips are uniformly sampled ",
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+ "text": "238 and we average the model predictions across these 10 views and report top-1 classification accuracy. \n239 For further implementation details, please refer to the supplemental material. ",
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+ "Figure 1: Ablation study of the proposed sampling policy on reducing the discrepancy measures. "
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+ "text": "240 4.2 Ablation Study ",
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+ "text": "In the following, we discuss multiple ablation studies to assess our main hypothesis that, a hierarchical sampling policy, as described in Section 3, enables better representations to be learned by increasing the portion of hard negative pairs which the contrastive loss function observes. Here, pretraining uses $90 \\%$ of either Movie or TV dataset, while the remaining $10 \\%$ constitute a held-out validation set7 on which we report the discrepancy measures. ",
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+ "text": "Sample size $( k )$ Figure 1a illustrates that compared to the baseline sampling denoted by $k = 1$ , our approach $k > 1 ,$ ) effectively shrinks the gap between $S ^ { - }$ and ${ \\mathcal { S } } ^ { \\neq }$ when measured either directly or against $S ^ { + }$ . Its pattern of behavior also perfectly follows our earlier intuition (ref. Section 3). In particular, given a fixed minibatch budget, a larger $k$ favors more training instances to be sampled from fewer number of long-form contents. That increases the portion of hard negative pairs, thus pushes the contrastive loss to more aggressively separate mismatched audio-video pairs from the same movie, which leads model to maintain less of the content-exclusive artifacts in the embedding space. In the most extreme case, $k = 6 4$ , all the training instances are sampled from the same movie. From Table 1, we observe that different variants of our sampling policy, with no imposed temporal constraint, i.e. $w = M _ { n }$ , outperform the baseline on all three downstream tasks ",
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+ "text": "Sampling window $\\mathbf { \\Pi } ^ { ( w ) }$ Smaller $w$ forces samples that belong to same movie to be drawn from a shorter temporal window, hence growing the probability that they look/sound very much alike (i.e. harder negative pairs). That is, it should further diminish the discrepancy measures. Figure 1b illustrates this behavior where we gradually increase $w$ while $k = 1 6$ . However, from Table 1, it does not seem that tuning for $w$ , i.e $w \\neq M _ { n }$ , provides a meaningful gain on downstream tasks. This implies that commonalities which persist throughout the duration of a movie are sufficiently powerful signals to be exploited for generating hard negatives. We hypothesize that different scenes both within and across different movies and TV shows are of variety of length, thus a fixed $w$ is sub-optimal. Ideally, we should identify scene boundaries and and dynamically modify $w$ during sampling, something which we leave for future iterations of this work. ",
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+ "text": "Temporally adjacent samples. Along the lines of previous observations, Figure 1c shows that indeed drawing temporally adjacent snippets from the same long-form content, i.e. $w = k$ , results in aggressively reducing the discrepancy measures. This behavior is agnostic with respect to $k$ yet exacerbates as $k$ grows. Note that, the contrastive loss is an instance-discrimination objective function. Therefore, forcing it to distinguish between temporally adjacent snippets, that naturally sound and look extremely similar, leaves no choice for the model but to discard valuable semantic notions, which predictably leads to poor representations, also confirmed by result reported in Table 1. ",
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+ "table_body": "<table><tr><td colspan=\"5\">pretraining dataset: Movie</td></tr><tr><td>k</td><td>w</td><td>HMDB51</td><td>ESC50</td><td>UCF101</td></tr><tr><td>1</td><td>1</td><td>60.32</td><td>86.50</td><td>85.69</td></tr><tr><td>4</td><td>Mn</td><td>61.37</td><td>89.91</td><td>85.38</td></tr><tr><td>8</td><td>Mn</td><td>62.09</td><td>88.75</td><td>86.06</td></tr><tr><td>16</td><td>Mn</td><td>62.92</td><td>88.33</td><td>86.30</td></tr><tr><td>32</td><td>Mn</td><td>61.04</td><td>88.00</td><td>85.98</td></tr><tr><td>64</td><td>Mn</td><td>61.30</td><td>86.83</td><td>85.43</td></tr><tr><td>16</td><td>64</td><td>60.26</td><td>87.00</td><td>83.61</td></tr><tr><td>16</td><td>128</td><td>60.58</td><td>86.50</td><td>85.30</td></tr><tr><td>16</td><td>256</td><td>62.02</td><td>87.75</td><td>84.85</td></tr><tr><td>16</td><td>512</td><td>61.30</td><td>87.08</td><td>85.38</td></tr><tr><td>16</td><td>1024</td><td>60.65</td><td>86.16</td><td>84.61</td></tr><tr><td>16</td><td>2048</td><td>61.83</td><td>87.66</td><td>85.11</td></tr><tr><td>4</td><td>4</td><td>60.19</td><td>88.00</td><td>84.66</td></tr><tr><td>16</td><td>16</td><td>56.86</td><td>88.75</td><td>82.71</td></tr><tr><td>64</td><td>64</td><td>57.45</td><td>84.58</td><td>82.68</td></tr><tr><td colspan=\"5\">pretraining dataset:TV</td></tr><tr><td>k</td><td>w</td><td>HMDB51</td><td>ESC50</td><td>UCF101</td></tr><tr><td>1</td><td>1</td><td>56.40</td><td>85.50</td><td>84.37</td></tr><tr><td>8</td><td>Mn</td><td>61.50</td><td>87.50</td><td>85.96</td></tr><tr><td>16</td><td>Mn</td><td>61.69</td><td>89.00</td><td>85.64</td></tr><tr><td>8</td><td>64</td><td>60.58</td><td>88.00</td><td>85.96</td></tr><tr><td>8</td><td>128</td><td>60.00</td><td>85.66</td><td>85.77</td></tr><tr><td>16</td><td>256</td><td>61.30</td><td>86.41</td><td>85.01</td></tr></table>",
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+ "text": "Movies vs. TV Shows. To confirm that our sampling policy behaves consistently across both movies and TV shows, Figure 1d illustrates the discrepancy measures computed on TV dataset. We observe similar effectiveness when using $k$ and $w$ as tuning nobs for reducing either ${ \\mathsf { K L } } ( S ^ { - } \\parallel S ^ { \\neq } )$ or the gap between ${ \\mathsf { K L } } ( S ^ { - } \\parallel S ^ { + } )$ and $\\mathsf { K L } ( S ^ { \\neq } \\parallel S ^ { + } )$ . Table 1 demonstrates that different variants of our approach significantly outperform the baseline, i.e. $k = 1$ . We attribute the larger gains achieved when using TV instead of Movie dataset to the fact that content diversity is naturally lower when pretraining on TV shows since each one includes many episodes that all are characterized with the same content-exclusive artifacts. ",
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+ "text": "Color jitter. We have established so far that commonalities which persist throughout the duration of a long-form content, things likely associated with color pallet, frequent appearance of the main character’s face and voice, and repetitive scenes can be exploited for learning better representations. That is, one may naturally assume that employing data augmentation techniques like color jitter should be helpful since by distorting content-exclusive visual artifacts, color jitter is expected to reduce ${ \\mathsf { K L } } ( S ^ { - } \\parallel S ^ { \\neq } )$ . Figure 2 illustrates the effect of color jitter, where brightness, contrast, and saturation jitter values are chosen uniformly from $\\mathtt { [ m a x ( 0 , 1 - \\sigma ) , 1 + \\sigma ] }$ . We observe that color jitter reduces the discrepancy measures for the baseline but not as much as it can be obtained by our proposed sampling policy $( k > 1 )$ ), and even then according to Table 2 only yields a slight gain on downstream tasks. ",
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+ "text": "$\\ell _ { 2 }$ -normalized feature space. The common practice [2, 31, 39, 7] is to compute contrastive loss in $\\ell _ { 2 }$ -normalized feature space, where according to [47] the temperature hyper-parameter, $\\tau$ , controls the strength of penalties on hard negative samples. We explored this with two widely-used $\\tau$ values. From Table 2, we observe that compared to operating in an unnormalized embedding space, adopting such design choice results in a large performance drop on HMDB51[23] while other downstream benchmarks see only negligible gains. ",
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526
+ "Table 4: Effect of self-supervised learning from curated versus uncurated data on different downstream tasks. The “years” column indicates the duration of the pretraining datasets in years. "
527
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+ "table_body": "<table><tr><td>method</td><td>pretraining dataset</td><td>uncurated</td><td>years</td><td>HMDB51</td><td>ESC50</td><td>UCF101</td></tr><tr><td>Ours</td><td>Movie</td><td></td><td>0.7</td><td>62.9</td><td>88.3</td><td>86.3</td></tr><tr><td>Ours</td><td>TV</td><td></td><td>0.7</td><td>61.7</td><td>89.0</td><td>85.6</td></tr><tr><td>XDC[3]</td><td>IG-Random16M</td><td>√</td><td>5</td><td>55.2</td><td>84.3</td><td>84.1</td></tr><tr><td>XDC[3]</td><td>IG-Random65M</td><td>√</td><td>21</td><td>61.2</td><td>86.3</td><td>88.8</td></tr><tr><td>XDC[3]</td><td>IG-Kinetics16M</td><td>×</td><td>5</td><td>57.3</td><td>82.5</td><td>87.6</td></tr><tr><td>XDC[3]</td><td>IG-Kinetics65M</td><td>×</td><td>21</td><td>63.1</td><td>84.8</td><td>91.5</td></tr></table>",
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+ "text": "Curated vs. Uncurated data. To the best of our knowledge, the only other uncurated dataset used for audio-visual self-supervised learning is IGRandom[3]8. Table 4 confirms that learning from uncurated movies and TV shows is extremely effective. Our results significantly exceed those of XDC[3] obtained on IG-Random16M despite using a simpler model and 7 times smaller volume of pretraining data. Even in comparison to IG-Random65M with 30 times larger data, we obtain better performances on 2 out of 3 benchmarks. The most promising of our findings though is how competitive our results are against XDC[3] when it is trained on variants of IG-Kinetics which are not only curated but also orders of magnitude larger. With all that, we confidently reject the notion that audio-visual self-supervised learning from uncurated data considerably lags behind utilizing large-scale curated datasets. ",
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+ "text": "Effect of genre. The distribution of genre among movies used in our pretraining is the closest we have to taxonomy in the curated datasets. So, it is worth examining the quality of our learned representations under various genre distributions. To do so, given a fixed pretraining budget $( N = 1 . 6 \\mathsf { K } )$ , we compare four different scenarios where movies used in the pretraining are distributed i) non-uniformly over all genres except Drama, and Comedies, ii) non-uniformly over Drama, and Comedies, iii) uniformly over all genres, and iv) non-uniformly over all genres. Table 3 confirms that indeed there is very little difference between the aforementioned setups when it comes to transfer learning to the downstream tasks. ",
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575
+ "Table 2: Effect of color jitter $( \\sigma )$ and computing contrastive loss in $\\ell _ { 2 }$ -normalized embedding space with temperature hyper-parameter $( \\tau )$ on different downstream tasks. "
576
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+ "table_body": "<table><tr><td>k</td><td>0</td><td>HMDB51</td><td>ESC50</td><td>UCF101</td></tr><tr><td>1</td><td>0.0</td><td>60.32</td><td>86.50</td><td>85.69</td></tr><tr><td>16</td><td>0.0</td><td>62.92</td><td>88.33</td><td>86.30</td></tr><tr><td>1</td><td>1.0</td><td>60.45</td><td>87.66</td><td>84.82</td></tr><tr><td>16</td><td>0.5</td><td>60.13</td><td>87.75</td><td>85.98</td></tr><tr><td>16</td><td>1.0</td><td>61.11</td><td>88.33</td><td>85.93</td></tr><tr><td>k</td><td>T</td><td>HMDB51</td><td>ESC50</td><td>UCF101</td></tr><tr><td>16</td><td>0.07</td><td>60.78</td><td>87.08</td><td>86.86</td></tr><tr><td>16</td><td>0.30</td><td>60.78</td><td>89.25</td><td>85.72</td></tr></table>",
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590
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591
+ "Table 3: Effect of genre distribution in Movie dataset on different downstream tasks. Experiments are conducted with input spatial resolution of $1 1 2 \\times 1 1 2$ pixels. "
592
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+ "table_body": "<table><tr><td>setting</td><td>HMDB51</td><td>ESC50</td><td>UCF101</td></tr><tr><td>i</td><td>57.58</td><td>86.50</td><td>82.44</td></tr><tr><td>ii</td><td>56.99</td><td>85.50</td><td>82.39</td></tr><tr><td>i</td><td>56.27</td><td>85.25</td><td>82.87</td></tr><tr><td>iv</td><td>56.40</td><td>86.75</td><td>83.24</td></tr></table>",
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+ "text": "4.3 Comparison with state-of-the-art ",
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+ "text": "Table 5 compares our proposed approach of learning from Movies and TV shows against the best performing audio-visual self-supervised learning methods. In general, our numbers are comparable with the best existing results reported in the literature, even with much less data and considerably simpler model/training procedure9. It is interesting that training on Movie dataset alone obtains ",
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640
+ "table_caption": [
641
+ "Table 5: Comparison with state-of-the-art. Dataset abbreviations: AudioSet[11], HowTo100M[28], IG-Kinetics65M [12]; their length in years is given in the “years” column. “Arch.” denotes the architecture of video backbone $( f )$ . [2]† indicates when the corresponding model use only audio and video, and not text modality. For a fair comparison, when using only Movie dataset, we train for twice as many epochs as our other variants in order to match their total number of gradient updates. "
642
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644
+ "table_body": "<table><tr><td>Method</td><td>Arch.</td><td>pretraining dataset</td><td>curated</td><td>years</td><td>HMDB51</td><td>UCF101</td><td>ESC50</td></tr><tr><td>GDT[39]</td><td>R(2+1)D-18</td><td>AS</td><td></td><td>1</td><td>66.1</td><td>92.5</td><td>88.5</td></tr><tr><td>GDT[39]</td><td>R(2+1)D-18</td><td>IG65M</td><td></td><td>21</td><td>72.8</td><td>95.2</td><td></td></tr><tr><td>XDC[3]</td><td>R(2+1)D-18</td><td>AS</td><td>√</td><td>1</td><td>61.0</td><td>91.2</td><td>84.8</td></tr><tr><td>XDC[3]</td><td>R(2+1)D-18</td><td>IG65M</td><td>√</td><td>21</td><td>67.4</td><td>94.2</td><td></td></tr><tr><td>AVTS[22]</td><td>MC3</td><td>AS</td><td></td><td>1</td><td>61.6</td><td>89.0</td><td>82.3</td></tr><tr><td>AVID[31]</td><td>R(2+1)D-18</td><td>AS</td><td></td><td>1</td><td>64.7</td><td>91.5</td><td>89.1</td></tr><tr><td>MMV[2]+</td><td>R(2+1)D-18</td><td>AS</td><td>√</td><td>1</td><td>70.1</td><td>91.5</td><td>85.6</td></tr><tr><td>MMV[2]+</td><td>S3D-G</td><td>AS</td><td>√</td><td>1</td><td>68.2</td><td>90.1</td><td>86.1</td></tr><tr><td>MMV[2]+</td><td>S3D-G</td><td>AS+HT</td><td>√</td><td>16</td><td>68.3</td><td>91.1</td><td>87.2</td></tr><tr><td>Ours (k=16)</td><td>R(2+1)D-18</td><td>Movie</td><td>X</td><td>0.7</td><td>64.5</td><td>87.9</td><td>88.8</td></tr><tr><td>Ours (k=8)</td><td>R(2+1)D-18</td><td>Movie+TV</td><td>X</td><td>1.4</td><td>65.0</td><td>87.7</td><td>89.1</td></tr><tr><td>Ours (k=16)</td><td>R(2+1)D-18</td><td>Movie+TV</td><td>X</td><td>1.4</td><td>65.1</td><td>88.5</td><td>89.1</td></tr><tr><td>Ours (k=32)</td><td>R(2+1)D-18</td><td>Movie+TV</td><td>X</td><td>1.4</td><td>65.6</td><td>88.7</td><td>88.2</td></tr></table>",
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+ "text": "358 comparable performance to the cases where both TV and Movie datasets are used for pretraining. \n359 This further confirms the richness of the training data which movies and TV shows can provide \n360 to self-supervised learning problems. We also see that increasing $k$ even beyond 8 gives further \n361 incremental gains on action recognition benchmarks. ",
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+ "text": "362 5 Conclusion ",
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+ "text": "363 \n364 \n365 \n366 \n367 \n368 \n369 \n370 \n371 \n372 \n373 ",
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+ "text": "Despite its amazing recent progress, state-of-the-art self-supervised learning still heavily relies on supervised, i.e. curated, large-scale datasets for pretraining. In this work, we have shown that pretraining solely on uncurated data in forms of movies and TV shows, even at a comparatively small scale, can give rise to representations which are capable of competing with the state-of-theart of more complex architectures trained on larger curated datasets. This comes contrary to the current literature which tends to suggest that learning from uncurated data largely falls behind the use of curated alternatives. We intentionally made design decisions to keep our approach and training strategy as simple as possible to demonstrate that learning decently powerful audio-visual representations does not necessarily require gigantic data and compute resources. Through extensive set of experiments, our work establishes for the first time the efficacy of self-supervised learning of audio-visual representations from movies and TV shows. ",
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+ "text": "374 6 Broader impact ",
701
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+ "text": "Potential benefits. Our work shows that competitive multimodal representations can be learned from a comparatively small volume of uncurated data in the form of movies and TV shows. Besides minimizing any sort of human-involvement, which we believe must have already been paid an extra attention to in the literature, our work demonstrates that one does not require gigantic data and compute resources for effective self-supervised pretraining. Such results promise a more democratized research arena where smaller groups are not alienated due lack of sufficient compute resources. More importantly, lowering the compute requirements naturally reduces any environmental effects which training these models can potentially have. ",
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+ "text": "Potential risks. Any machine learning method is susceptible to the potential underlying biases in the data. This is more important for self-supervised methods that deal with huge volumes, often not evaluated by diverse group of humans for any fairness concerns. The same is generally true in our case which requires us to make sure that titles that are included in training are diverse and inclusive. ",
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Zitnick, and M. Hebert. Shuffle and learn: unsupervised learning using temporal order verification. In European Conference on Computer Vision, pages 527–544. Springer, 2016. [31] P. Morgado, N. Vasconcelos, and I. Misra. Audio-visual instance discrimination with cross-modal agreement. arXiv preprint arXiv:2004.12943, 2020. \n63 [32] V. Nair and G. E. Hinton. Rectified linear units improve restricted boltzmann machines. In ICML, 2010. \n4 [33] M. Noroozi and P. Favaro. Unsupervised learning of visual representations by solving jigsaw puzzles. In European Conference on Computer Vision, pages 69–84. Springer, 2016. \n6 [34] M. Noroozi, H. Pirsiavash, and P. Favaro. Representation learning by learning to count. In Proceedings of the IEEE International Conference on Computer Vision, pages 5898–5906, 2017. \n68 [35] A. v. d. Oord, Y. Li, and O. Vinyals. Representation learning with contrastive predictive coding. arXiv preprint arXiv:1807.03748, 2018. [36] A. Owens and A. A. Efros. Audio-visual scene analysis with self-supervised multisensory features. In Proceedings of the European Conference on Computer Vision (ECCV), pages 631–648, 2018. [37] A. Owens, J. Wu, J. H. McDermott, W. T. Freeman, and A. Torralba. Ambient sound provides supervision for visual learning. In European conference on computer vision, pages 801–816. Springer, 2016. [38] D. Pathak, P. Krahenbuhl, J. Donahue, T. Darrell, and A. A. Efros. Context encoders: Feature learning by inpainting. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2536–2544, 2016. [39] M. Patrick, Y. M. Asano, R. Fong, J. F. Henriques, G. Zweig, and A. Vedaldi. Multi-modal self-supervision from generalized data transformations. arXiv preprint arXiv:2003.04298, 2020. [40] K. J. Piczak. Esc: Dataset for environmental sound classification. In Proceedings of the 23rd ACM international conference on Multimedia, pages 1015–1018, 2015. [41] S. Purushwalkam and A. Gupta. Demystifying contrastive self-supervised learning: Invariances, augmentations and dataset biases. In H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 3407–3418, 2020. \n[42] K. Soomro, A. R. Zamir, and M. Shah. Ucf101: A dataset of 101 human actions classes from videos in the wild. arXiv preprint arXiv:1212.0402, 2012. \n[43] C. Sun, A. Shrivastava, S. Singh, and A. Gupta. Revisiting unreasonable effectiveness of data in deep learning era. In Proceedings of the IEEE international conference on computer vision, pages 843–852, 2017. \n[44] Y. Tian, D. Krishnan, and P. Isola. Contrastive multiview coding. arXiv preprint arXiv:1906.05849, 2019. \n[45] D. Tran, H. Wang, L. Torresani, J. Ray, Y. LeCun, and M. Paluri. A closer look at spatiotemporal convolutions for action recognition. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pages 6450–6459, 2018. \n[46] C. Vondrick, A. Shrivastava, A. Fathi, S. Guadarrama, and K. Murphy. Tracking emerges by colorizing videos. In Proceedings of the European conference on computer vision (ECCV), pages 391–408, 2018. \n[47] F. Wang and H. Liu. Understanding the behaviour of contrastive loss. arXiv preprint arXiv:2012.09740, 2020. \n[48] J. Wang, J. Jiao, L. Bao, S. He, Y. Liu, and W. Liu. Self-supervised spatio-temporal representation learning for videos by predicting motion and appearance statistics. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 4006–4015, 2019. \n[49] D. Xu, J. Xiao, Z. Zhao, J. Shao, D. Xie, and Y. Zhuang. Self-supervised spatiotemporal learning via video clip order prediction. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 10334–10343, 2019. \n[50] R. Zhang, P. Isola, and A. A. Efros. Colorful image colorization. In European conference on computer vision, pages 649–666. Springer, 2016. ",
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+ "text": "Xu He, Herbert Jaeger \nDepartment of Computer Science and Electrical Engineering \nJacobs University Bremen \nBremen, 28759, Germany \n{x.he,h.jaeger}@jacobs-university.de ",
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+ "text": "Catastrophic interference has been a major roadblock in the research of continual learning. Here we propose a variant of the back-propagation algorithm, “conceptor-aided backprop” (CAB), in which gradients are shielded by conceptors against degradation of previously learned tasks. Conceptors have their origin in reservoir computing, where they have been previously shown to overcome catastrophic forgetting. CAB extends these results to deep feedforward networks. On the disjoint and permuted MNIST tasks, CAB outperforms two other methods for coping with catastrophic interference that have recently been proposed. ",
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+ "text": "1 INTRODUCTION ",
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+ "text": "Agents with general artificial intelligence are supposed to learn and perform well on multiple tasks. Continual learning refers to the scenarios where a machine learning system can retain previously acquired skills while learning new ones. However, when trained on a sequence of tasks, neural networks usually forget about previous tasks after their weights are adjusted for a new task. This notorious problem known as catastrophic interference (CI) (McCloskey & Cohen, 1989; Ratcliff, 1990; French, 1999; Kumaran et al., 2016) poses a serious challenge towards continual learning. ",
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+ "text": "Many approaches have been proposed to overcome or mitigate the problem of CI in the last three decades (Hinton & Plaut, 1987; French, 1991; Ans & Rousset, 1997; French, 1997; Srivastava et al., 2014). Especially recently, an avalanche of new methods in the deep learning field has brought about dramatic improvements in continual learning in neural networks. Kirkpatrick et al. (2017) introduced a regularization-based method called elastic weight consolidation (EWC), which uses the posterior distribution of parameters for the old tasks as a prior for the new task. They approximated the posterior by a Gaussian distribution with the parameters for old tasks as the mean and the inverse diagonal of the Fisher information matrix as the variance. Lee et al. (2017) introduced two incremental moment matching (IMM) methods called mean-IMM and mode-IMM. Mean-IMM approximates the distribution of parameters for both old and new tasks by a Gaussian distribution, which is estimated by minimizing its KL-divergence from the mixture of two Gaussian posteriors, one for the old task and the other one for the new task. Mode-IMM estimates the mode of this mixture of two Gaussians and uses it as the optimal parameters for both tasks. ",
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+ "text": "In the field of Reservoir Computing (Jaeger, 2001; Maass et al., 2002), an effective solution to CI using conceptors was proposed by Jaeger (2014) to incrementally train a recurrent neural network to generate spatial-temporal signals. Conceptors are a general-purpose neuro-computational mechanism that can be used in a diversity of neural information processing tasks including temporal pattern classification, one-shot learning, human motion pattern generation, de-noising and signal separation (Jaeger, 2017). In this paper, we adopt and extend the method introduced in Jaeger (2014) and propose a conceptor-aided backpropagation (CAB) algorithm to train feed-forward networks. For each layer of a network, CAB computes a conceptor to characterize the linear subspace spanned by the neural activations in that layer that have appeared in already learned tasks. When the network is trained on a new task, CAB uses the conceptor to adjust the gradients given by backpropagation so that the linear transformation restricted to the characterized subspace will be preserved after the gradient descent procedure. Experiment results of two benchmark tests showed highly competitive performance of CAB. ",
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+ "text": "The rest of this paper is structured as follows. Section 2 introduces conceptors and their application to incremental learning by ridge regression. Section 3 extends the method to stochastic gradient descent and describes the CAB algorithm. Section 4 compares its performance on the permuted and disjoint MNIST tasks to recent methods that address the same problem. Finally we conclude our paper in Section 5. ",
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+ "text": "2 INCREMENTAL RIDGE REGRESSION BY CONCEPTORS ",
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+ "text": "This section reviews the basics of conceptor theory and its application to incrementally training linear readouts of recurrent neural networks as used in reservoir computing. A comprehensive treatment can be found in (Jaeger, 2014). ",
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+ "text": "2.1 CONCEPTORS ",
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+ "Figure 1: 3D point clouds (black dots) and their corresponding conceptors, represented by ellipsoids whose axes are the singular vectors of conceptors and the lengths of these axes match the singular values of conceptors. Each edge of the plot boxes range from $- 1$ to $+ 1$ admitted by neural dynamics with a tanh nonlinearity; conceptor ellipsiods lie inside the unit sphere. "
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+ "text": "In brief, a matrix conceptor $C$ for some vector-valued random variable $\\boldsymbol { x } \\in \\mathbb { R } ^ { N }$ is defined as a linear transformation that minimizes the following loss function. ",
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+ "text": "$$\n\\mathbb { E } _ { x } [ | | x - C x | | ^ { 2 } ] + \\alpha ^ { - 2 } | | C | | _ { \\mathrm { f r o } } ^ { 2 }\n$$",
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+ "text": "where $\\alpha$ is a control parameter called aperture and $| | \\cdot | | _ { \\mathrm { f r o } }$ is the Frobenius norm. This optimization problem has a closed-form solution ",
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+ "text": "$$\nC = R ( R + \\alpha ^ { - 2 } I ) ^ { - 1 }\n$$",
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+ "text": "where $R = \\mathbb { E } _ { x } [ x x ^ { \\top } ]$ is the $N \\times N$ correlation matrix of $x$ , and $I$ is the $N \\times N$ identity matrix. This result given in (2) can be understood by studying the singular value decomposition (SVD) of $C$ . If $R = \\bar { U } \\Sigma U ^ { \\top }$ is the SVD of $R$ , then the SVD of $C$ is given as $U S U ^ { \\top }$ , where the singular values $s _ { i }$ of $C$ can be written in terms of the singular values $\\sigma _ { i }$ of $R$ : $s _ { i } = \\sigma _ { i } / ( \\sigma _ { i } + \\alpha ^ { - 2 } ) \\mathbf { \\bar { \\Omega } } \\in [ 0 , 1 )$ . In intuitive terms, $C$ is a soft projection matrix on the linear subspace where the samples of $x$ lie. For a vector $y$ in this subspace, $C$ acts like the identity: $C y \\approx y$ , and when some noise $\\epsilon$ orthogonal to the subspace is added to $y$ , $C$ de-noises: $C ( y + \\epsilon ) \\approx y$ . Figure 1 shows the ellipsoids corresponding to three sets of $\\mathbb { R } ^ { 3 }$ points. We define the quota $Q ( C )$ of a conceptor to be the mean singular values: $\\begin{array} { r } { Q ( C ) : = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } s _ { i } } \\end{array}$ . Intuitively, the quota measures the fraction of the total dimensions of the entire vector space that is claimed by $C$ . ",
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+ "text": "Moreover, logic operations that satisfy most laws of Boolean logic can be defined on matrix conceptors as the following: ",
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+ "text": "$$\n\\begin{array} { c } { { \\neg C : = I - C , } } \\\\ { { C ^ { i } \\vee C ^ { j } : = ( R ^ { i } + R ^ { j } ) ( R ^ { i } + R ^ { j } + \\alpha ^ { - 2 } I ) ^ { - 1 } } } \\\\ { { C ^ { i } \\wedge C ^ { j } : = \\neg ( \\neg C ^ { i } \\vee \\neg C ^ { j } ) } } \\end{array}\n$$",
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+ "text": "where $\\lnot C$ softly projects onto a linear subspace that can be roughly understood as the orthogonal complement of the subspace characterized by $C$ . $C ^ { i } \\vee C ^ { j }$ is the conceptor computed from the union of the two sets of sample points from which $C ^ { i }$ and $C ^ { j }$ are computed. It describes a space that is approximately the sum of linear subspaces characterized by $C ^ { i }$ and $C ^ { j }$ , respectively. The definition of $\\bar { C } ^ { i } \\wedge C ^ { j }$ reflects de Morgan’s law. Figure 2 illustrates the geometry of these operations. ",
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+ "Figure 2: Geometry of Boolean operations on 2-dimensional conceptors. The OR (resp. AND) operation gives a conceptor whose ellipsoid approximately is the smallest (largest) ellipsoid enclosing (contained in) the argument conceptor’s ellipsoids. "
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+ "text": "2.2 INCREMENTAL RIDGE REGRESSION ",
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+ "text": "This subsection explains how conceptors can be applied to master continual learning in a simple linear model trained on a supervised task by ridge regression. The training is done sequentially on multiple input-to-output mapping tasks. This simplified scenario illustrates the working principle of continual learning with conceptors and will later be used repeatedly as a sub-procedure in the CAB algorithm for training multilayer feed-forward networks. ",
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+ "text": "Consider a sequence of $m$ incoming tasks indexed by $j$ . We denote the training dataset for the $j$ -th task by $\\{ ( x _ { 1 } ^ { j } , y _ { 1 } ^ { j } ) , \\cdot \\cdot \\cdot , ( x _ { n } ^ { j } , y _ { n } ^ { j } ) \\}$ , where $\\boldsymbol { x } _ { i } ^ { j } \\in \\mathbb { R } ^ { N }$ are input vectors and $y _ { i } ^ { j } \\in \\mathbb { R } ^ { M }$ their corresponding target outputs. Whenever the training dataset for a new task is available, the incremental learning method will compute a matrix conceptor $C ^ { j }$ for the input variable of the new task using Equation 2 and update the linear model, resulting in a sequence of linear models $W ^ { 1 } , \\ldots W ^ { m }$ such that $W ^ { j }$ solves not only the $j$ -th task but also all previous tasks: for $k \\leq j , y ^ { k } \\approx W ^ { j } x ^ { k }$ . The conceptor $C ^ { j }$ is a soft projection matrix onto the linear subspace spanned by input patterns from the $j$ -th task. Then, $A ^ { j - 1 } \\overset { \\cdot } { = } \\overset { \\cdot } { C } ^ { 1 } \\vee \\cdots \\vee C ^ { j - 1 }$ characterizes the memory space already claimed by the tasks $1 , \\ldots , j - 1$ and $F ^ { j } = \\neg A ^ { j - 1 }$ , the orthogonal complement of $A ^ { j } - 1$ , represents the memory space still free for the $j$ -th task. Here “memory space” refers to the linear space of input vectors. In detail, this method proceeds in the following way: ",
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+ "text": "• Initialization (no task trained yet): $W ^ { 0 } = 0 _ { M \\times N } , A ^ { 0 } = 0 _ { N \\times N }$ • Incremental task learning: For tasks $j = 1 , \\ldots , m$ do: ",
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+ "text": "1. Store the input vectors from the $j$ -th training dataset of size $n$ into a $N \\times n$ sized input collection matrix $X ^ { j }$ , and store the output vectors into a $M \\times n$ sized output collection matrix $Y ^ { j }$ . ",
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+ "text": "2. Compute the conceptor for this task by $C ^ { j } ~ = ~ R ^ { j } ( R ^ { j } + \\alpha ^ { - 2 } I ) ^ { - 1 }$ , where $R ^ { j } \\ =$ $\\scriptstyle { \\frac { 1 } { n } } X ^ { j ^ { \\prime } } X ^ { j ^ { \\top } }$ ",
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+ "text": "3. Train an increment matrix $W _ { i n c } ^ { j }$ (to be added to $W ^ { j - 1 }$ , yielding $W ^ { j }$ ), with the crucial aid of a helper conceptor $F ^ { j }$ : ",
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+ "text": "(a) $F ^ { j } : = \\neg A ^ { j - 1 }$ (comment: this conceptor characterizes the “still disposable” memory space for the $j$ -th task), \n(b) $T : = Y ^ { j } - ( W ^ { j - 1 } X ^ { j } )$ (comment: this matrix consists of target values for a linear regression to compute $W _ { i n c . } ^ { j }$ ), \n(c) $S : = F ^ { j } X ^ { j }$ (comment: this matrix consists of input arguments for the linear regression), ",
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+ "text": "(d) $W _ { i n c } ^ { j } \\ : = \\ : ( ( S S ^ { \\top } / n + \\lambda ^ { - 2 } I ) ^ { - 1 } S T ^ { \\top } / n ) ^ { \\top }$ (comment: carry out the regression, regularized by $\\lambda ^ { - 2 }$ ), ",
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+ "text": "$W ^ { j }$ : $W ^ { j } = W ^ { j - 1 } + W _ { i n c } ^ { j }$ ",
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+ "text": "5. Update $A : A ^ { j } = A ^ { j - 1 } \\vee C ^ { j }$ (comment: this is possible due to the associativity of the ∨ operation on conceptors) ",
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+ "text": "The weight increment $W _ { i n c } ^ { j }$ does not interfere much with the previously learned weights $W ^ { j - 1 }$ because the regularization in step 3(d) constrains the row space of $W _ { i n c } ^ { j }$ to be only the linear subspace spanned by input arguments defined in 3(c), which are inside the kernel of $W ^ { j - 1 }$ due to the projection by $F ^ { j }$ . Intuitively speaking, when learning a new task, this algorithm exploits only the components of input vectors in the still unused space (kernel of $W ^ { j - 1 }$ , characterized by $F ^ { j }$ ) to compensate errors for the new task and leaves the directions in the already used memory space (row space of $W ^ { j - 1 }$ , characterized by $A ^ { j - 1 }$ ) intact. ",
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+ "text": "3 CONCEPTOR-AIDED SGD AND BACK-PROP ",
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+ "text": "In this section, we first derive a stochastic gradient descent version of the algorithm described in the previous section, then present the procedure of CAB. ",
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+ "text": "3.1 SGD ",
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+ "text": "In the algorithm introduced in the previous section, $W _ { i n c } ^ { j }$ is computed by ridge regression, which offers a closed-form solution to minimize the following cost function ",
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+ "img_path": "images/bb210187a9e9e8a9bad53192c659d369c6f15fd1146b743001d1faffcd1fd527.jpg",
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+ "text": "$$\n\\mathcal { I } ( W _ { i n c } ^ { j } ) : = \\mathbb { E } [ | W _ { i n c } ^ { j } s - t | ^ { 2 } ] + \\lambda ^ { - 2 } | W _ { i n c } ^ { j } | _ { \\mathrm { f r o } } ^ { 2 }\n$$",
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+ "text": "where $t = y ^ { j } - W ^ { j - 1 } x ^ { j } , s = F ^ { j } x ^ { j }$ . One can also minimize this cost function by stochastic gradient descent (SGD), which starts from an initial guess of $W _ { i n c } ^ { j }$ and repeatedly performs the following update ",
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+ "text": "$$\nW _ { i n c } ^ { j } W _ { i n c } ^ { j } - \\eta \\nabla _ { W _ { i n c } ^ { j } } \\mathcal { I } ( W _ { i n c } ^ { j } )\n$$",
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+ "text": "where $\\eta$ is the learning rate and the gradient is given by: ",
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+ "text": "$$\n\\nabla _ { W _ { i n c } ^ { j } } \\mathcal { I } ( W _ { i n c } ^ { j } ) = 2 \\mathbb { E } [ ( W _ { i n c } ^ { j } s - t ) s ^ { \\top } ] + 2 \\lambda ^ { - 2 } W _ { i n c } ^ { j }\n$$",
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+ "text": "Substituting $t$ by $y ^ { j } - W ^ { j - 1 } x ^ { j }$ and $s$ by $F ^ { j } x ^ { j } = ( I - A ^ { j - 1 } ) x ^ { j }$ in (8), we get ",
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+ "img_path": "images/4c41edc9e7dfcf8bce7d76a83ce326b5e6cc672a68c647cd4027359168eb7d49.jpg",
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+ "text": "$$\n\\begin{array} { r l } & { \\nabla _ { W _ { i n c } ^ { j } } \\mathcal { I } ( W _ { i n c } ^ { j } ) = 2 \\mathbb { E } [ ( W _ { i n c } ^ { j } ( I - A ^ { j - 1 } ) x ^ { j } - y ^ { j } + W ^ { j - 1 } x ^ { j } ) s ^ { \\top } ] + 2 \\lambda ^ { - 2 } W _ { i n c } ^ { j } } \\\\ & { \\phantom { \\frac { 1 } { 1 } } = 2 \\mathbb { E } [ ( - W _ { i n c } ^ { j } A ^ { j - 1 } x ^ { j } + ( W ^ { j - 1 } + W _ { i n c } ^ { j } ) x ^ { j } - y ^ { j } ) s ^ { \\top } ] + 2 \\lambda ^ { - 2 } W _ { i n c } ^ { j } } \\end{array}\n$$",
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+ "text": "Due to the regularization term in the cost function, as the optimization goes on, eventually $W _ { i n c }$ will null the input components that are not inside the linear subspace characterized by $F ^ { j }$ , hence $W _ { i n c } ^ { j } A ^ { j - 1 } x ^ { j }$ will converge to 0 as the algorithm proceeds. In addition, since $W ^ { j } = W ^ { j - 1 } + W _ { i n c } ^ { j }$ , (10) can be simplified to ",
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+ "text": "$$\n\\nabla _ { { W _ { i n c } ^ { j } } } \\mathcal { I } ( W _ { i n c } ^ { j } ) = 2 \\mathbb { E } [ ( W ^ { j } x ^ { j } - y ^ { j } ) s ^ { \\top } ] + 2 \\lambda ^ { - 2 } W _ { i n c } ^ { j }\n$$",
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+ "text": "Adding $W ^ { j - 1 }$ to both sides of (7), we obtain the update rule for $W ^ { j }$ : ",
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+ "text": "$$\n\\begin{array} { r } { W ^ { j } W ^ { j } - 2 \\eta \\mathbb { E } [ e s ^ { \\top } ] + 2 \\eta \\lambda ^ { - 2 } W _ { i n c } ^ { j } } \\end{array}\n$$",
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+ "text": "where $e : = W ^ { j } x ^ { j } - y ^ { j }$ . In practice, at every iteration, the expected value can be approximated by a mini-batch of size $n _ { B }$ , indexed by $i _ { B }$ : ",
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+ "text": "$$\n\\hat { \\mathbb { E } } [ e s ^ { \\top } ] = \\frac { 1 } { n _ { B } } \\sum _ { i _ { B } = 0 } ^ { L } ( W ^ { j } x _ { i _ { B } } ^ { j } - y _ { i _ { B } } ^ { j } ) ( F ^ { j } x _ { i _ { B } } ^ { j } ) ^ { \\top } = \\frac { 1 } { n _ { B } } \\sum _ { i _ { B } = 0 } ^ { L } ( W ^ { j } x _ { i _ { B } } ^ { j } - y _ { i _ { B } } ^ { j } ) x _ { i _ { B } } ^ { j ^ { \\top } } F ^ { j ^ { \\top } }\n$$",
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+ "text": "where the transpose for $F ^ { j }$ can be dropped since it is symmetric. ",
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+ "text": "If we only train the $j$ −th task without considering the previous tasks, the update rule given by normal SGD is ",
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+ "img_path": "images/cbfb723fdc13449ace8b3e661d0f66dd28d6eac4cd41ad18f014438be577f008.jpg",
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+ "text": "$$\nW ^ { j } W ^ { j } - 2 \\eta \\mathbb { E } [ e x ^ { j \\top } ] + 2 \\eta \\lambda ^ { - 2 } W ^ { j }\n$$",
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+ "text": "Comparing this to the update rule in (12), we notice two modifications when a conceptor is adopted to avoid CI: first, the gradient of weights are calculated using the conceptor-projected input vector $s = F ^ { j } x ^ { j }$ instead of the original input vector $x ^ { j }$ ; second, regularization is done on the weight increment $W _ { i n c } ^ { j }$ rather than the final weight $W ^ { j }$ . These two modifications lead to our design of the conceptor-aided algorithm for training multilayer feed-forward networks. ",
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+ "text": "The basic idea of CAB is to guide the gradients of the loss function on every linear component of the network by a matrix conceptor computed from previous tasks during error back-propagation (Rumelhart et al., 1986), repeatedly applying the conceptor-aided SGD technique introduced in the previous section in every layer. ",
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+ "text": "Consider a feed-forward network with $L + 1$ layers, indexed by $l = 0 , \\ldots L$ , such that the 0-th and the $L$ -th layers are the input and output layers respectively. $W ^ { ( l ) }$ represents the linear connections between the $( l - 1 )$ -th and the $l$ -th layer, where we refer to the former as the pre-synaptic layer with respect to $W ^ { ( l ) }$ , and to the latter as the post-synaptic layer. We denote by $N ^ { ( l ) }$ the size of the $l$ -th layer (excluding the bias unit) and $A ^ { ( l ) ^ { j } }$ a conceptor characterizing the memory space in the $l$ -th layer used up by the first $j$ tasks. Let $\\sigma ( \\cdot )$ be the activation function of the nonlinear neurons and $\\theta$ all the parameters of the network to be trained. Then the incremental training method with CAB proceeds as follows: ",
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+ "text": "• Initialization (no task trained yet): $\\forall l = 0 , \\ldots , L - 1$ , $A ^ { ( l ) ^ { 0 } } : = 0 _ { ( N ^ { ( l ) } + 1 ) \\times ( N ^ { ( l ) } + 1 ) }$ , and randomly initialize $W ^ { ( l + 1 ) ^ { 0 } }$ to be a matrix of size $\\boldsymbol { N } ^ { ( l + 1 ) } \\times \\left( \\boldsymbol { N } ^ { ( l ) } + 1 \\right)$ . ",
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+ "text": "• Incremental task learning: For $j = 1 , \\ldots , m$ do: ",
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+ "text": "1. $\\forall l = 0 , \\ldots , L - 1 , F ^ { ( l ) ^ { j } } = \\lnot A ^ { ( l ) ^ { ( j - 1 ) } }$ . (This conceptor characterizes the still disposable vector space in layer l for learning task $j$ ) ",
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+ "text": "2. Update the network parameters $\\theta ^ { ( j - 1 ) }$ obtained after training the first $j - 1$ tasks to $\\theta ^ { j }$ by stochastic gradient descent, where the gradients are computed by CAB instead of the classical backprop. Algorithms 1 and 2 detail the forward and backward pass of CAB, respectively. Different from classical backprop, the gradients are guided by a matrix conceptor $F ^ { ( l ) ^ { j } }$ , such that in each layer only the activity in the still disposable memory space will contribute to the gradient. Note that the conceptors remain the same until convergence of the network for task $j$ . ",
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+ "text": "3. After training on the vectors, indexed by $i _ { B }$ $j$ -th task, run the forward procedure again on a batch of , taken from the $j$ -th training dataset, to collect activations $n _ { B }$ input $h _ { i _ { B } } ^ { ( l ) ^ { j } }$ of each layer into a $N ^ { ( l ) } \\times n _ { B }$ sized matrix $H ^ { ( l ) ^ { j } }$ , and set the correlation matrix $\\begin{array} { r } { R ^ { ( l ) ^ { j } } = \\frac { \\dot { 1 } } { n _ { B } } H ^ { ( l ) ^ { j } } ( H ^ { ( l ) ^ { j } } ) ^ { \\top } } \\end{array}$ . ",
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+ "text": "4. Compute a conceptor on the $l$ -th layer for the $j$ -th pattern by $C ^ { ( l ) ^ { j } } = R ^ { ( l ) ^ { j } } ( R ^ { ( l ) ^ { j } } +$ $\\alpha ^ { - 2 } \\hat { I } _ { N ^ { ( l ) } \\times N ^ { ( l ) } } ) ^ { - 1 } , \\forall l = 0 , \\dots , L \\bar { - } 1$ . Finding an optimal aperture can be done by a cross-validation search1. ",
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+ "text": "5. Update the conceptor for already used space in every layer: $A ^ { ( l ) ^ { j } } ~ = ~ A ^ { ( l ) ^ { j } } ~ \\vee$ $C ^ { ( l ) ^ { j } } , \\forall l = 0 , \\dots , L - 1$ . ",
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+ "text": "Algorithm 1 The forward procedure of conceptor-aided backprop, adapted from the traditional backprop. Input vectors are passed through a feed-forward network to compute the cost function. $\\mathcal { L } ( \\hat { y } ^ { j } , y ^ { j } )$ denotes the loss for the $j$ -th task, to which a regularizer $\\Omega ( \\theta _ { i n c } ^ { j } ) = \\Omega ( \\theta ^ { j } - \\theta ^ { j - 1 } ) =$ $| | \\theta ^ { j } - \\theta ^ { j - 1 } | | _ { \\mathrm { f r o } } ^ { 2 }$ is added to obtain the total cost $\\mathcal { I }$ , where $\\theta$ contains all the weights (biases are considered as weights connected to the bias units). The increment of parameters rather than the parameters themselves are regularized, similar to the conceptor-aided SGD. ",
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+ "text": "Require: Network depth, $l$ \nRequire: $W _ { . } ^ { ( l ) ^ { j } } , l \\in \\bar { \\{ 1 , \\ldots , L \\} }$ , the weight matrices of the network \nRequire: $x ^ { j }$ , one input vector of the $j$ -th task \nRequire: $y ^ { j }$ , the target output for $x ^ { j }$ \n1: $h ^ { ( 0 ) } = x ^ { j }$ \n2: for $l = 1 , \\dots L$ do 3: $b ^ { ( l ) } = [ h ^ { ( l - 1 ) \\top } , 1 ] ^ { \\top }$ , include the bias unit 4: $a ^ { ( l ) } = W ^ { ( l ) ^ { j } } b ^ { ( l ) }$ 5: $h ^ { ( l ) } = \\sigma ( a ^ { ( l ) } )$ \n6: end for \n7: ${ \\hat { y } } ^ { j } = h ^ { ( l ) }$ \n8: $\\mathcal { I } = \\mathcal { L } ( \\hat { y } ^ { j } , y ^ { j } ) + \\lambda \\Omega ( \\theta _ { i n c } ^ { j } )$ ",
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+ "text": "Algorithm 2 The backward procedure of conceptor-aided backprop for the $j$ -th task, adapted from the traditional backprop. The gradient $g$ of the loss function $\\mathcal { L }$ on the activations $a ^ { ( l ) }$ represents the error for the linear transformation $W ^ { ( l ) ^ { j } }$ between the $( l - 1 )$ -th and the l−th layers. In the standard backprop algorithm, the gradient of $\\mathcal { L }$ on $W ^ { ( l ) ^ { j } }$ is computed as an outer product of the post-synaptic errors $g$ and the pre-synaptic activities $\\boldsymbol { h } ^ { ( l - 1 ) }$ . This resembles the computation of the gradient in the linear SGD algorithm, which motivates us to apply conceptors in a similar fashion as in the conceptor-aided SGD. Specifically, we project the gradient $\\nabla _ { W ^ { ( l ) } } j \\mathcal { L }$ by the matrix conceptor F (l−1) that indicates the free memory space on the pre-synaptic layer. ",
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+ "text": "1: ",
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+ "text": "$$\n\\boldsymbol { g } \\gets \\nabla _ { \\boldsymbol { \\hat { y } } } \\mathcal { I } = \\nabla _ { \\boldsymbol { \\hat { y } } } \\mathcal { L } ( \\boldsymbol { \\hat { y } } , \\boldsymbol { y } )\n$$",
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+ "text": "2: for $l = L , L - 1 , \\ldots , 1$ do ",
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+ "text": "3: Convert the gradient on the layer’s output into a gradient on the pre-nonlinearity activation ( $\\odot$ denotes element-wise multiplication): ",
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+ "img_path": "images/e51e4c07de07bc3a0aa0c3d64b116800014e165d213d39e017b507defac78ecc.jpg",
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+ "text": "$$\ng \\nabla _ { a ^ { ( l ) } } \\mathcal { I } = g \\odot \\sigma ^ { \\prime } ( a ^ { ( l ) } )\n$$",
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+ "text": "4: Compute the gradient of weights, project it by $F ^ { ( l - 1 ) ^ { j } }$ , and add it to the regularization term on the increment: ",
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+ "text": "$$\n\\begin{array} { l } { { \\nabla _ { W ^ { ( l ) ^ { j } } } \\mathcal { I } = g \\big ( F ^ { ( l - 1 ) ^ { j } } b ^ { ( l - 1 ) } \\big ) ^ { \\top } + \\lambda \\nabla _ { W ^ { ( l ) ^ { j } } } \\Omega \\big ( \\theta _ { i n c } ^ { j } \\big ) = g b ^ { ( l - 1 ) ^ { \\top } } F ^ { ( l - 1 ) ^ { j } } + 2 \\lambda W _ { i n c } ^ { ( l ) ^ { j } } } } \\\\ { { \\quad \\quad = g b ^ { ( l - 1 ) ^ { \\top } } F ^ { ( l - 1 ) ^ { j } } + 2 \\lambda \\big ( W ^ { ( l ) ^ { j } } - W ^ { ( l ) ^ { j - 1 } } \\big ) } } \\end{array}\n$$",
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+ "text": "5: Propagate the gradients w.r.t. the next lower-level hidden layers activations: ",
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+ "img_path": "images/39249121bc8988dc293256d4ed97d0b288629980d9f4b2ec49ec40d73ecd163b.jpg",
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+ "text": "$$\ng \\gets \\nabla _ { h ^ { ( l - 1 ) } } \\mathcal { I } = W ^ { ( l ) ^ { j } } { } ^ { \\top } g\n$$",
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+ "text": "6: end for ",
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+ "image_caption": [
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+ "Figure 3: Average performance across already learned permuted MNIST tasks using CAB or EWC "
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+ "text": "4 EXPERIMENTS ",
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+ "text": "4.1 PERMUTED MNIST EXPERIMENT ",
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+ "text": "To test the performance of CAB, we evaluated it on the permuted MNIST experiment (Srivastava et al., 2013; Goodfellow et al., 2014; Kirkpatrick et al., 2017; Lee et al., 2017), where a sequence of pattern recognition tasks are created from the MNIST dataset (LeCun et al., 1998). For each task, a random permutation of input image pixels is generated and applied to all images in MNIST to obtain a new shuffled dataset, equally difficult to recognize as the original one, the objective of each task is to recognize these images with shuffled pixels. ",
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+ "text": "For a proof-of-concept demonstration, we trained a simple but sufficient feed-forward network with [784-100-10] of neurons to classify 10 permuted MNIST datasets. The network has logistic sigmoid neurons in both hidden and output layers, and is trained with mean squared error as the cost function. Vanilla SGD was used in all experiments to optimize the cost function. Learning rate and aperture were set to 0.1 and 4, respectively. For comparison, we also tested EWC on the same task with the same network architecture, based on the implementation by Seff (2017). The parameters chosen for the EWC algorithm were 0.01 for the learning rate and 15 for the weight of the Fisher penalty term. Figure 3 shows the performance of CAB on this task, the average testing accuracy is $9 5 . 2 \\%$ after learning all 10 tasks sequentially. Although a fair amount of effort was spent on searching for optimal parameters for EWC, the accuracies shown here might still not reflect its best performance. However, the same experiment with EWC was also conducted in Kemker et al. (2017), where the authors reimplemented EWC on a network with higher capacity (2 hidden layers and 400 ReLU neurons per layer) and the resulting average accuracy after learning 10 tasks sequentially was shown to be around $93 \\%$ . ",
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+ {
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+ "type": "text",
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+ "text": "Since all tasks are generated by permuting the same dataset, the portion of the input space occupied by each of them should have the same size. However, as more tasks are learned, the chance that the space of a new task will overlap with the already used input space increases. Figure 4 shows the singular value spectra and quota of the input and hidden layer conceptors every time after a new task is learned. As the incremental learning proceeds, it becomes less likely for a new task to be in the free space. For example, the second task increases the quota of the input layer memory space by 0.1, whereas the 10th task increases it by only 0.03. However, CAB still manages to make the network learn new tasks based on their input components in the non-overlapping space. ",
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+ {
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+ "image_caption": [
1005
+ "(b) Singular value spectra of conceptors $A ^ { ( 1 ) ^ { j } }$ on the hidden layer. ",
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+ "Figure 4: The development of singular value spectra of conceptors for “used-up” space on the input layer and hidden layer during incremental learning of 10 permuted MNIST tasks. Quota of these conceptors are displayed in the legends. "
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+ "text": "4.2 DISJOINT MNIST EXPERIMENT ",
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+ "text": "We then applied CAB to categorize the disjoint MNIST datasets into 10 classes (Srivastava et al., 2013; Lee et al., 2017). In this experiment, the original MNIST dataset is divided into two disjoint datasets with the first one consisting of data for the first five digits (0 to 4), and the second one of the remaining five digits (5 to 9). This task requires a network to learn these two datasets one after the other, then examines its performance of classifying the entire MNIST testing images into 10 classes. The current state-of-the-art accuracy on this task, averaged over 10 learning trials, is $9 4 . 1 2 ( \\pm 0 . 2 7 ) \\%$ , achieved by Lee et al. (2017) using IMM. They also tested EWC on the same task and the average accuracy was $5 2 . 7 2 ( \\pm 1 . 3 6 ) \\%$ . ",
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+ {
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+ "text": "To test our method, we trained a feed-forward network with [784-800-10] neurons. Logistic sigmoid nonlinearities were used in both hidden and output layers, and the network was trained with vanilla SGD to minimize mean squared errors. The aperture $\\alpha = 9$ was used for all conceptors on all layers, learning rate $\\eta$ and regularization coefficient $\\lambda$ were chosen to be 0.1 and 0.005 respectively. The accuracy of CAB on this task, measured by repeating the experiment 10 times, is $9 4 . 9 1 ( \\pm 0 . { \\dot { 3 } } 0 ) \\%$ . It is worth mentioning that the network used by Lee et al. (2017) for testing IMM and EWC had [784-800-800-10] rectified linear units (ReLU), so CAB achieved better performance with fewer layers and neurons. ",
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+ "text": "4.3 COMPUTATIONAL COST ",
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+ {
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+ "text": "If a conceptor is computed by ridge regression, the time complexity is $O ( n N ^ { 2 } + N ^ { 3 } )$ when the design matrix is dense, where $n$ is the number of samples and $N$ the number of features. In terms of wall time measures, the time taken to compute a conceptor from the entire MNIST training set (in this case, $n = 5 5 0 0 0$ images and $N = 7 8 4$ pixels, corresponding to the input layer in our networks) is 0.42 seconds of standard notebook CPU time on average. Although we did not implement it in these experiments, incremental online adaptation of conceptors by gradient descent is also possible in principle and would come at a cost of $\\dot { O ( N ^ { 2 } ) }$ per update. ",
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+ "text": "5 CONCLUSION ",
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+ "text": "In this work, we first reviewed the conceptor-based incremental ridge regression algorithm, introduced in section 3.11 of Jaeger (2014) for memory management in recurrent neural networks. Then we derived its stochastic gradient descent version for optimizing the same objective. Finally we designed a conceptor-aided backprop algorithm by applying a conceptor to every linear layer of a feed-forward network. This method uses conceptors to guide gradients of parameters during the backpropagation procedure. As a result, learning a new task interferes only minimally with previously learned tasks, and the amount of already used network capacity can be monitored via the singular value spectra and quota of conceptors. ",
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+ "type": "text",
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+ "text": "In Jaeger (2014), different scenarios for continual learning are investigated in a reservoir computing setting. Two extreme cases are obtained when (i) the involved learning tasks are entirely unrelated to each other, versus (ii) all tasks come from the same parametric family of learning tasks. The two cases differ conspicuously with regards to the geometry of involved conceptors, and with regards to opportunities to re-use previously acquired functionality in subsequent learning episodes. The permuted MNIST task is an example of (i) while the disjoint MNIST task rather is of type (ii). Conceptors provide an analytical tool to discuss the “family relatedness” and enabling/disabling conditions for continual learning in geometrical terms. Ongoing and future research is devoted to a comprehensive mathematical analysis of these phenomena which in our view lie at the heart of understanding continual learning. ",
1100
+ "bbox": [
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+ ],
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+ "page_idx": 8
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+ },
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+ {
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+ "type": "text",
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+ "text": "ACKNOWLEDGMENTS ",
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+ "text_level": 1,
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+ "text": "David E Rumelhart, Geoffrey E Hinton, and Ronald J Williams. Learning representations by backpropagating errors. Nature, 323:533–535, 1986. \nAri Seff. Implementation of overcoming catastrophic forgetting in neural networks in tensorflow. GitHub Repository, 2017. https://github.com/ariseff/ overcoming-catastrophic. \nRupesh K Srivastava, Jonathan Masci, Sohrob Kazerounian, Faustino Gomez, and Jurgen Schmid- ¨ huber. Compete to compute. In Advances in Neural Information Processing Systems, pp. 2310– 2318, 2013. http://papers.nips.cc/paper/5059-compete-to-compute.pdf. \nVipin Srivastava, Suchitra Sampath, and David J Parker. Overcoming catastrophic interference in connectionist networks using Gram-Schmidt orthogonalization. PloS ONE, 9(9):e105619, 2014. ",
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1
+ # ROBUST AND GENERALIZABLE VISUAL REPRESENTATION LEARNING VIA RANDOM CONVOLUTIONS
2
+
3
+ Zhenlin $\mathbf { X } \mathbf { u } ^ { 1 }$ , Deyi $\mathbf { L i u } ^ { 1 }$ , Junlin Yang2, Colin Raffel1, and Marc Niethammer1
4
+
5
+ 1 University of North Carolina at Chapel Hill 2 Yale University 1{zhenlinx, mn, craffel}@cs.unc.edu, deyi@live.unc.edu 2junlin.yang@yale.edu
6
+
7
+ # ABSTRACT
8
+
9
+ While successful for various computer vision tasks, deep neural networks have shown to be vulnerable to texture style shifts and small perturbations to which humans are robust. In this work, we show that the robustness of neural networks can be greatly improved through the use of random convolutions as data augmentation. Random convolutions are approximately shape-preserving and may distort local textures. Intuitively, randomized convolutions create an infinite number of new domains with similar global shapes but random local texture. Therefore, we explore using outputs of multi-scale random convolutions as new images or mixing them with the original images during training. When applying a network trained with our approach to unseen domains, our method consistently improves the performance on domain generalization benchmarks and is scalable to ImageNet. In particular, in the challenging scenario of generalizing to the sketch domain in PACS and to ImageNet-Sketch, our method outperforms state-of-art methods by a large margin. More interestingly, our method can benefit downstream tasks by providing a more robust pretrained visual representation. 1
10
+
11
+ # 1 INTRODUCTION
12
+
13
+ Generalizability and robustness to out-of-distribution samples have been major pain points when applying deep neural networks (DNNs) in real world applications (Volpi et al., 2018). Though DNNs are typically trained on datasets with millions of training samples, they still lack robustness to domain shift, small perturbations, and adversarial examples (Luo et al., 2019). Recent research has shown that neural networks tend to use superficial features rather than global shape information for prediction even when trained on large-scale datasets such as ImageNet (Geirhos et al., 2019). These superficial features can be local textures or even patterns imperceptible to humans but detectable to DNNs, as is the case for adversarial examples (Ilyas et al., 2019). In contrast, image semantics often depend more on object shapes rather than local textures. For image data, local texture differences are one of the main sources of domain shift, e.g., between synthetic virtual images and real data (Sun & Saenko, 2014). Our goal is therefore to learn visual representations that are invariant to local texture and that generalize to unseen domains. While texture and color may be treated as different concepts, we follow the convention in Geirhos et al. (2019) and include color when talking about texture.
14
+
15
+ We address the challenging setting of robust visual representation learning from single domain data. Limited work exists in this setting. Proposed methods include data augmentation (Volpi et al., 2018; Qiao et al., 2020; Geirhos et al., 2019), domain randomization (Tobin et al., 2017; Yue et al., 2019), self-supervised learning (Carlucci et al., 2019), and penalizing the predictive power of low-level network features (Wang et al., 2019a). Following the spirit of adding inductive bias towards global shape information over local textures, we propose using random convolutions to improve the robustness to domain shifts and small perturbations. While recently Lee et al. (2020) proposed a similar technique for improving the generalization of reinforcement learning agents in unseen environments, we focus on visual representation learning and examine our approach on visual domain generalization benchmarks. Our method also includes the multiscale design and a mixing variant. In addition, considering that many computer vision tasks rely on training deep networks based on ImageNet-pretrained weights (including some domain generalization benchmarks), we ask “Can a more robust pretrained model make the finetuned model more robust on downstream tasks?” Different from (Kornblith et al., 2019; Salman et al., 2020) who studied the transferability of a pretrained ImageNet representation to new tasks while focusing on in-domain generalization, we explore generalization performance on unseen domains for new tasks.
16
+
17
+ ![](images/fd7c9f9d8b200815c39951855989c43207583b7b966425745cfdd84427e984b8.jpg)
18
+ Figure 1: Top: Illustration that RandConv randomize local texture but preserve shapes in the image. Middle: First column is the input image of size $2 2 4 ^ { 2 }$ ; following columns are convolutions results using random filters of different sizes $k$ . Bottom: Mixing results between an image and one of its random convolution results with different mixing coefficients $\alpha$ .
19
+
20
+ We make the following contributions:
21
+
22
+ • We develop RandConv, a data augmentation technique using multi-scale random-convolutions to generate images with random texture while maintaining global shapes. We explore using the RandConv output as training images or mixing it with the original images. We show that a consistency loss can further enforce invariance under texture changes. • We provide insights and justification on why RandConv augments images with different local texture but the same semantics with the shape-preserving property of random convolutions. We validate RandConv and its mixing variant in extensive experiments on synthetic and realworld benchmarks as well as on the large-scale ImageNet dataset. Our methods outperform single domain generalization approaches by a large margin on digit recognition datasets and for the challenging case of generalizing to the Sketch domain in PACS and to ImageNet-Sketch. • We explore if the robustness/generalizability of a pretrained representation can transfer. We show that transferring a model pretrained with RandConv on ImageNet can further improve domain generalization performance on new downstream tasks on the PACS dataset.
23
+
24
+ # 2 RELATED WORK
25
+
26
+ Domain Generalization (DG) aims at learning representations that perform well when transferred to unseen domains. Modern techniques range between feature fusion (Shen et al., 2019), metalearning (Li et al., 2018a; Balaji et al., 2018), and adversarial training (Shao et al., 2019; Li et al., 2018b). Note that most current DG work (Ghifary et al., 2016; Li et al., 2018a;b) requires a multisource training setting to work well. However, in practice, it might be difficult and expensive to collect data from multiple sources, such as collecting data from multiple medical centers (Raghupathi & Raghupathi, 2014). Instead, we consider the more strict single-domain generalization DG setting, where we train the model on source data from a single domain and generalize it to new unseen domains (Carlucci et al., 2019; Wang et al., 2019b).
27
+
28
+ Domain Randomization (DR) was first introduced as a DG technique by Tobin et al. (2017) to handle the domain gap between simulated and real data. As the training data in (Tobin et al., 2017) is synthesized in a virtual environment, it is possible to generate diverse training samples by randomly selecting background images, colors, lighting, and textures of foreground objects. When a simulation environment is not accessible, image stylization can be used to generate new domains (Yue et al., 2019; Geirhos et al., 2019). However, this requires extra effort to collect data and to train an additional model; further, the number of randomized domains is limited by the number of predefined styles.
29
+
30
+ Data Augmentation has been widely used to improve the generalization of machine learning models (Simard et al., 2003). DR approaches can be considered a type of synthetic data augmentation. To improve performance on unseen domains, Volpi et al. (2018) generate adversarial examples to augment the training data; Qiao et al. (2020) extend this approach via meta-learning. As with other adversarial training algorithms, significant extra computation is required to obtain adversarial examples.
31
+
32
+ Learning Representations Biased towards Global Shape Geirhos et al. (2019) demonstrated that convolutional neural networks (CNNs) tend to use superficial local features even when trained on large datasets. To counteract this effect, they proposed to train on stylized ImageNet, thereby forcing a network to rely on object shape instead of textures. Wang et al. improved out-of-domain performance by penalizing the correlation between a learned representation and superficial features such as the gray-level co-occurrence matrix (Wang et al., 2019b), or by penalizing the predictive power of local, low-level layer features in a neural network via an adversarial classifier (Wang et al., 2019a). Our approach shares the idea that learning representations invariant to local texture helps generalization to unseen domains. However, RandConv avoids searching over many hyper-parameters, collecting extra data, and training other networks. It also scales to large-scale datasets since it adds minimal computation overhead.
33
+
34
+ Random Mapping in Machine Learning Random projections have also been effective for dimensionality reduction based on the distance-preserving property of the Johnson–Lindenstrauss lemma (Johnson & Lindenstrauss, 1984). (Vinh et al., 2016) applied random projections on entire images as data augmentation to make neural networks robust to adversarial examples. Lee et al. (2020) recently used random convolutions to help reinforcement learning (RL) agents generalize to new environments. Neural networks with fixed random weights can encode meaningful representations (Saxe et al., 2011) and are therefore useful for neural architecture search (Gaier & Ha, 2019), generative models (He et al., 2016b), natural language processing (Wieting & Kiela, 2019), and RL (Osband et al., 2018; Burda et al., 2019). In contrast, RandConv uses non-fixed randomly-sampled weights to generate images with different local texture.
35
+
36
+ # 3 RANDCONV: RANDOMIZE LOCAL TEXTURE AT DIFFERENT SCALES
37
+
38
+ We propose using a convolution layer with non-fixed random weights as the first layer of a DNN during training. This strategy generates images with random local texture but consistent shapes, and is beneficial for robust visual representation learning. Sec. 3.1 justifies the shape-preserving property of a random convolution layer. Sec. 3.2 describes RandConv, our data augmentation algorithm using a multi-scale randomized convolution layer and input mixing.
39
+
40
+ # 3.1 A RANDOM CONVOLUTION LAYER PRESERVES GLOBAL SHAPES
41
+
42
+ Convolution is the key building block for deep convolutional neural networks. Consider a convolution layer with filters $\pmb { \Theta } \doteq \mathbb { R } ^ { h \times w \times C _ { i n } \times C _ { o u t } }$ with an input image $\mathbf { I } \in \mathbb { R } ^ { H \times W \times C _ { i n } }$ , where $H$ and $W$ are the height and width of the input and $C _ { i n }$ and $C _ { o u t }$ are the number of feature channels for the input and output, and $h$ and $w$ are the height and width of the layer’s filter. The output (with appropriate input padding) will be $\mathbf { g } = \mathbf { I } * \mathbf { \Theta } \Theta$ with $\mathbf { g } \in \mathbb { R } ^ { H \times W \times C _ { o u t } }$ .
43
+
44
+ In images, nearby pixels with similar color or texture can be grouped into primitive shapes that represent parts of objects or the background. A convolution layer linearly projects local image patches to features at corresponding locations on the output map using shared parameters. While a convolution with random filters can project local patches to arbitrary output features, the output of a random linear projection approximately preserves relative similarity between input patches, proved in Appendix B. In other words, since any two locations within the same shape have similar local textures in the input image, they tend to be similar in the output feature map. Therefore, shapes that emerge in the output feature map are similar to shapes in the input image provided that the filter size is sufficiently small compared to the size of a typical shape.
45
+
46
+ In other words, the size of a convolution filter determines the smallest shape it can preserve. For example, 1x1 random convolutions preserve shapes at the single-pixel level and thus work as a random color mapping; large filters perturb shapes smaller than the filter size that are considered local texture of a shape at this larger scale. See Fig. 1 for examples. More discussion and a formal proof are in Appendix A and $B$ .
47
+
48
+ # Algorithm 1 Learning with Data Augmentation by Random Convolutions
49
+
50
+ 1: Input: Model $\Phi$ , task loss $\mathcal { L } _ { t a s k }$ , training images $\{ I _ { i } \} _ { i = 1 } ^ { N }$ and their labels $\{ y _ { i } \} _ { i = 1 } ^ { N }$ , pool of filter sizes
51
+ $\mathcal { K } = \{ 1 , . . . , n \}$ , fraction of original data $p$ , whether to mix with original images, consistency loss weight $\lambda$
52
+ 2: function RA N DCO N V(I, $\kappa$ , mix, $p$ )
53
+ 3: Sample $p _ { 0 } \sim U ( 0 , 1 )$
54
+ 4: if $p _ { 0 } < p$ and mix is False then
55
+ 5: return $I$ . When not in mix mode, use the original image with probability $p$
56
+ 6: else
57
+ 7: 8: Sample scale Sample conv $k \sim \kappa$ weights $\Theta \in \mathbb { R } ^ { k \times k \times 3 \times 3 } \sim N ( 0 , \frac { 1 } { 3 k ^ { 2 } } )$
58
+ 9: $I _ { r c } \bar { = } I * \Theta$ . Apply convolution on $I$
59
+ 10: if mix is True then
60
+ 11: Sample $\alpha \sim U ( 0 , 1 )$
61
+ 12: return $\alpha I + ( 1 - \alpha ) I _ { r c }$ . Mix with original images
62
+ 13: else
63
+ 14: return $I _ { r c }$
64
+ 15: Learning Objective:
65
+ 16: for $i = 1 N$ do
66
+ 17: for $j = 1 3$ do
67
+ 18: $\hat { y } _ { i } ^ { j } = \Phi ( \mathrm { R a n d C o n v } ( I _ { i } ) )$ . Predict labels for three augmented variants of the same image
68
+ 19: $\begin{array} { r } { \mathcal { L } _ { c o n s } = \lambda \sum _ { j = 1 } ^ { 3 } \mathrm { K L } ( \hat { y } _ { i } ^ { j } | | \bar { y } _ { i } ) } \end{array}$ where $\textstyle { \bar { y } } _ { i } = \sum _ { j = 1 } ^ { 3 } { \hat { y } } _ { i } ^ { j } / 3$ . Consistency Loss
69
+ 20: $\mathcal { L } = \mathcal { L } _ { t a s k } ( \hat { y } _ { i } ^ { 1 } , y _ { i } ) + \lambda \mathcal { L } _ { c o n s }$ . Learning with the task loss and the consistency loss
70
+
71
+ Sec. 3.1 discussed how outputs of randomized convolution layers approximately maintain shape information at a scale larger than their filter sizes. Here, we develop our RandConv data augmentation technique using a randomized convolution layer with $C _ { o u t } = C _ { i n }$ to generate shape-consistent images with randomized texture (see Alg. 1). Our goal is not to use RandConv to parameterize or represent texture as in previous filter-bank based texture models (Heeger & Bergen, 1995; Portilla & Simoncelli, 2000). Instead, we only use the three-channel outputs of RandConv as new images with the same shape and different “style” (loosely referred to as "texture"). We also note that, a convolution layer is different from a convolution operation in image filtering. Standard image filtering applies the same 2D filter on three color channels separately. In contrast, our convolution layer applies three different $3 D$ filters and each takes all color channels as input and generates one channel of the output. Our proposed RandConv variants are as follows:
72
+
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+ $\mathbf { R C _ { i m g } }$ : Augmenting Images with Random Texture A simple approach is to use the randomized convolution layer outputs, $I * \Theta$ , as new images; where $\Theta$ are the randomly sampled weights and $I$ is a training image. If the original training data is in the domain $D ^ { 0 }$ , a sampled weight $\Theta _ { k }$ generates images with consistent global shape but random texture forming the random domain $D ^ { k }$ . Thus, by random weight sampling, we obtain an infinite number of random domains $D ^ { 1 } , D ^ { 1 } , \ldots , D ^ { \infty }$ . Input image intensities are assumed to be a standard normal distribution $N ( 0 , 1 )$ (which is often true in practice thanks to data whitening). As the outputs of RandConv should follow the same distribution,√ we sample the convolution weights from $N ( 0 , \sigma ^ { 2 } )$ where $\sigma = 1 / \sqrt { C _ { i n } \times h \times w }$ , which is commonly applied for network initialization (He et al., 2015). We include the original images for training at a ratio $p$ as a hyperparameter.
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+ $\mathbf { R C _ { m i x } }$ : Mixing Variant As shown in Fig. 1, outputs from $\mathrm { R C } _ { \mathrm { i m g } }$ can vary significantly from the appearance of the original images. Although generalizing to domains with significantly different local texture distributions is useful, we may not want to sacrifice much performance on domains similar to the training domain. Inspired by the AugMix (Hendrycks et al., 2020b) strategy, we propose to blend the original image with the outputs of the RandConv layer via linear convex combinations $\alpha I + ( 1 - \alpha { \bar { ) } } ( I * \Theta )$ , where $\alpha$ is the mixing weight uniformly sampled from $[ 0 , 1 ]$ .In $\operatorname { R C } _ { \operatorname* { m i x } }$ , the RandConv outputs provide shape-consistent perturbations of the original images. Varying $\alpha$ , we continuously interpolate between the training domain and the randomly sampled domains of $\mathtt { R C } _ { \mathtt { i m g } }$
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+ Multi-scale Texture Corruption As discussed in Sec. 3.1„ image shape information at a scale smaller than a filter’s size will be corrupted by RandConv. Therefore, we can use filters of varying sizes to preserve shapes at various scales. We choose to uniformly randomly sample a filter size $k$ from a pool $\mathcal { K } = 1 , 3 , . . . n$ before sampling convolution weights $\pmb { \Theta } \in \mathbb { R } ^ { k \times k \times C _ { i n } ^ { \bot } \times C _ { o u t } }$ from a Gaussian distribution $\begin{array} { r } { N ( 0 , \frac { 1 } { k ^ { 2 } C _ { i n } } ) } \end{array}$ . Fig. 1 shows examples of multi-scale RandConv outputs.
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+ Consistency Regularization To learn representations invariant to texture changes, we use a loss encouraging consistent network predictions for the same RandConv-augmented image for different random filter samples. Approaches for transform-invariant domain randomization (Yue et al., 2019), data augmentation (Hendrycks et al., 2020b), and semi-supervised learning (Berthelot et al., 2019) use similar strategies. We use Kullback-Leibler (KL) divergence to measure consistency. However, enforcing prediction similarity of two augmented variants may be too strong. Instead, following (Hendrycks et al., 2020b), we use RandConv to obtain 3 augmentation samples of image $I$ : $G _ { j } =$ $\mathtt { R a n d C o n v } ^ { j } ( I )$ for $j = 1 , 2 , 3$ and obtain their predictions with a model $\Phi$ : $y ^ { j } = \Phi ( G ^ { j } )$ . We then compute the relaxed loss as $\textstyle \lambda \sum _ { j = 1 } ^ { 3 } \mathrm { K L } ( y ^ { j } | | \bar { y } )$ , where $\textstyle { \bar { y } } = \sum _ { j = 1 } ^ { 3 } y ^ { j } / 3$ is the sample average.
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+ # 4 EXPERIMENTS
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+ Secs. 4.1 to 4.3 evaluate our methods on the following datasets: multiple digit recognition datasets, PACS, and ImageNet-sketch. Sec. 4.4 uses PACS to explore the out-of-domain generalization of a pretrained representation in transfer learning by checking if pretraining on ImageNet with our method improves the domain generalization performance in downstream tasks. All experiments are in the single-domain generalization setting where training and validation sets are drawn from one domain. Additional experiments with ResNet18 as the backbone are given in the Appendix.
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+ # 4.1 DIGIT RECOGNITION
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+ The five digit recognition datasets (MNIST (LeCun et al., 1998), MNIST-M (Ganin et al., 2016), SVHN (Netzer et al., 2011), SYNTH (Ganin & Lempitsky, 2014) and USPS (Denker et al., 1989)) have been widely used for domain adaptation and generalization research (Peng et al., 2019a;b; Qiao et al., 2020). Following the setups in (Volpi et al., 2018) and (Qiao et al., 2020), we train a simple CNN with 10,000 MNIST samples and evaluate the accuracy on the test sets of the other four datasets. We also test on MNIST-C (Mu & Gilmer, 2019), a robustness benchmark with 15 common corruptions of MNIST and report the average accuracy over all corruptions.
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+ ![](images/286ba736abe267b08e4fb701d0feca1de7966d9429056f4bfd681517880dd19b.jpg)
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+ Figure 2: Average accuracy and 5-run variance of MNIST model on MNIST-M, SVHN, SYNTH and USPS. Studies for: (a) original data fraction $p$ for $\mathrm { R C } _ { \mathrm { i m g } }$ ; (b) multiscale design (1-n refers to using scales 1,3,..,n) for $\mathrm { R C } _ { \mathrm { i m g } , p = 0 . 5 }$ (orange) and $\mathrm { R C } _ { \mathrm { m i x } }$ (blue); (c) consistency loss weight $\lambda$ for $\mathrm { R C } _ { \mathrm { i m g 1 - 7 } , p = 0 . 5 }$ (orange) and $\mathrm { R C } _ { \mathrm { m i x 1 - 7 } }$ (blue).
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+ Selecting Hyperparameters and Ablation Study. Fig. 2(a) shows the effect of the hyperparameter $p$ on $\mathrm { R C } _ { \mathrm { i m g } }$ with filter size 1. We see that adding only $1 0 \%$ RandConv data $( p = 0 . 9 )$ ) immediately improves the average performance (DG-Avg) on MNIST-M, SVHN, SYNTH and USPS performance from 53.53 to 69.19, outperforming all other approaches (see Tab. 1) for every dataset. We choose $p = 0 . 5$ , which obtains the best DG-Avg. Fig. 2(b) shows results for a multiscale ablation study. Increasing the pool of filter sizes up to 7 improves DG-Avg performance. Therefore we use multiscale 1-7 to study the consistency loss weight $\lambda$ , shown in Fig. 2(c). Adding the consistency loss improves both RandConv variants on DG-avg: $\mathrm { R C } _ { \mathrm { m i x 1 - 7 } }$ favors $\lambda = 1 0$ while $\mathrm { R C } _ { \mathrm { i m g 1 - 7 } , p = 0 . 5 }$ performs similarly for $\lambda = 5$ and $\lambda = 1 0$ . We choose $\lambda = 1 0$ for all subsequent experiments.
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+ Results. Tab. 1 compares the performance of $\operatorname { R C } _ { \operatorname* { i m g 1 - 7 } , p = 0 . 5 , \lambda = 1 0 }$ and $\mathrm { R C } _ { \mathrm { m i x 1 - 7 } , \lambda = 1 0 }$ with other state-of-the-art approaches. We show results of the adversarial training based methods GUD (Volpi et al., 2018), M-ADA (Qiao et al., 2020), and PAR (Wang et al., 2019a). The baseline model is trained only on the standard classification loss. To show RandConv is more than a trivial color/contrast adjustment method, we also compare to ColorJitter2 data augmentation (which randomly changes image brightness, contrast, and saturation) and GreyScale (where images are transformed to greyscale for training and testing). We also tested data augmentation with a fixed Laplacian of Gaussian filter (Band-Pass) of size $^ { = 3 }$ and $\sigma = 1$ and the data augmentation pipeline (Multi-Aug) that was used in a recently proposed large scale study on domain generalization algorithms and datasets (Gulrajani & Lopez-Paz, 2020). RandConv and its mixing variant outperforms the best competing method (MADA) by $17 \%$ on DG-Avg and achieves the best $9 1 . 6 2 \%$ accuracy on MNIST-C. While the difference between the two variants of RandConv is marginal, $\mathrm { R C } _ { \mathrm { m i x 1 - 7 } , \lambda = 1 0 }$ performs better on both DG-Avg and MNIST-C. When combined with Multi-Aug, RandConv achieves improved performance except on MNIST-C. Fig 3 shows t-SNE image feature plots for unseen domains generated by the baseline approach and $\mathrm { R C } _ { \mathrm { m i x 1 - 7 } , \lambda = 1 0 }$ . The RandConv embeddings suggest better generalization to unseen domains.
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+ Table 1: Average accuracy and 5-run standard deviation (in parenthesis) of MNIST10K model on MNIST-M, SVHN, SYNTH, USPS and their average (DG-avg); and average accuracy of 15 types of corruptions in MNIST-C. Both RandConv variants significantly outperform all other methods.
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+ <table><tr><td></td><td>MNIST</td><td>MNIST-M</td><td>SVHN</td><td>USPS</td><td>SYNTH</td><td>DG-Avg</td><td>MNIST-C</td></tr><tr><td>Baseline</td><td>98.40(0.84)</td><td>58.87(3.73)</td><td>33.41(5.28)</td><td>79.27(2.70)</td><td>42.43(5.46)</td><td>53.50(4.23)</td><td>88.20(2.10)</td></tr><tr><td>GreyScale</td><td>98.82(0.02)</td><td>58.41(0.99)</td><td>36.06(1.48)</td><td>80.45(1.00)</td><td>45.00(0.80)</td><td>54.98(0.86)</td><td>89.15(0.44)</td></tr><tr><td>ColorJitter</td><td>98.72(0.05)</td><td>62.72(0.66)</td><td>39.61(0.88)</td><td>79.18(0.60)</td><td>46.40(0.34)</td><td>56.98(0.39)</td><td>89.48(0.18)</td></tr><tr><td>BandPass</td><td>98.65(0.11)</td><td>70.22(2.73)</td><td>48.34(2.56)</td><td>78.60(0.82)</td><td>57.17(2.01)</td><td>63.58(1.89)</td><td>87.89(0.68)</td></tr><tr><td>MultiAug</td><td>98.80(0.05)</td><td>62.32(0.66)</td><td>39.07(0.68)</td><td>79.31(1.02)</td><td>46.48(0.80)</td><td>56.79(0.34)</td><td>89.54(0.11)</td></tr><tr><td>PAR (our imp)</td><td>98.79(0.05)</td><td>61.16(0.21)</td><td>36.08(1.27)</td><td>79.95(1.18)</td><td>45.48(0.35)</td><td>55.67(0.33)</td><td>89.34(0.45)</td></tr><tr><td>GUD</td><td>-</td><td>60.41</td><td>35.51</td><td>77.26</td><td>45.32</td><td>54.62</td><td>=</td></tr><tr><td>M-ADA</td><td>-</td><td>67.94</td><td>42.55</td><td>78.53</td><td>48.95</td><td>59.49</td><td></td></tr><tr><td>RCimg1-7, p=0.5, λ=5</td><td>98.86(0.05)</td><td>87.67(0.37)</td><td>54.95(1.90)</td><td>82.08(1.46)</td><td>63.37(1.58)</td><td>72.02(1.15)</td><td>90.94(0.51)</td></tr><tr><td>RCmix1-7,λ=10</td><td>98.85(0.04)</td><td>87.76(0.83)</td><td>57.52(2.09)</td><td>83.36(0.96)</td><td>62.88(0.78)</td><td>72.88(0.58)</td><td>91.62(0.77)</td></tr><tr><td>RCmix1-7,λ=10 + MultiAug</td><td>98.82(0.06)</td><td>87.89(0.29)</td><td>62.07(0.62)</td><td>84.39(1.02)</td><td>63.90(0.63)</td><td>74.56(0.46)</td><td>91.40(0.93)</td></tr></table>
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+ # 4.2 PACS EXPERIMENTS
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+ The PACS dataset (Li et al., 2018b) considers 7-class classification on 4 domains: photo, art painting, cartoon, and sketch, with very different texture styles. Most recent domain generalization work studies the multi-source domain setting on PACS and uses domain labels of the training data. Although we follow the convention to train on 3 domains and to test on the fourth, we simply pool the data from the 3 training domains as in (Wang et al., 2019a), without using domain labels during the training.
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+ Baseline and State-of-the-Art. Following (Li et al., 2017), we use Deep-All as the baseline, which finetunes an ImageNet-pretrained AlexNet on 3 domains using only the classification loss and tests on the fourth domain. We test our RandConv variants $\mathrm { R C } _ { \mathrm { i m g 1 - 7 } , p = 0 . 5 }$ and $\mathrm { R C } _ { \mathrm { m i x 1 - 7 } }$ with and without consistency loss, and ColorJitter/GreyScale/BandPass/MultiAug data augmentation as in the digit datasets. We also implemented PAR (Wang et al., 2019a) using our baseline model. $\mathrm { R C } _ { \mathrm { m i x 1 - 7 } }$ combined with MultiAug is also tested. Further, we compare to the following state-of-the-art approaches: Jigen (Carlucci et al., 2019) using self-supervision, MLDG (Li et al., 2018a) using meta-learning, and the conditional invariant deep domain generalization method CIDDG (Li et al., 2018c). Note that previous methods used different Deep-All baselines which make the final accuracy not directly comparable, and MLDG and CIDDG use domain labels for training.
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+ Figure 3: t-SNE feature embedding visualization for digit datasets for models trained on MNIST without (top) and with our $\mathrm { R C } _ { \mathrm { m i x 1 - 7 } , \lambda = 1 0 }$ approach (bottom). Different colors denote different classes.
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+ Table 2: Mean and 5-run standard deviation (in parenthesis) results for domain generalization on PACS. Best results with our Deep-All baseline are in bold. The domain name in each column represents the target domain. Base column indicates different baselines and results under different baselines are not directly comparable. MLDG and CIDDF used domain labels for training.
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+ <table><tr><td>Base</td><td>Method</td><td>Photo</td><td>Art</td><td>Cartoon</td><td>Sketch</td><td>Average</td></tr><tr><td rowspan="9">Ours</td><td>Deep-All</td><td>86.77(0.42) 83.93(1.47)</td><td>60.11(1.33) 61.60(1.18)</td><td>64.12(0.32) 62.12(0.61)</td><td>55.28(4.71) 60.07(2.47)</td><td>66.57(1.36) 66.93(0.83)</td></tr><tr><td>GreyScale</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ColorJitter</td><td>84.61(0.83)</td><td>59.01(0.24)</td><td>61.43(0.68)</td><td>62.44(1.68)</td><td>66.88(0.33)</td></tr><tr><td>BandPass</td><td>87.08(0.57)</td><td>59.46(0.27)</td><td>64.39(0.51)</td><td>55.39(2.95)</td><td>66.58(0.73)</td></tr><tr><td>MultiAug</td><td>85.21(0.47)</td><td>59.51(0.38)</td><td>62.88(1.01)</td><td>61.67(0.76)</td><td>67.32(0.23)</td></tr><tr><td>PAR (our imp.)</td><td>87.21(0.42)</td><td>60.17(0.95)</td><td>63.63(0.88)</td><td>55.83(2.57)</td><td>66.71(0.58)</td></tr><tr><td>RCimg1-7, p=0.5</td><td>86.50(0.72)</td><td>61.10(0.38)</td><td>64.24(0.62)</td><td>68.50(1.83)</td><td>70.09(0.43)</td></tr><tr><td>RCmix1-7</td><td>86.60(0.67)</td><td>61.74(0.90)</td><td>64.05(0.66)</td><td>69.74(0.66)</td><td>70.53(0.25)</td></tr><tr><td>RCmix1-7 + MultiAug</td><td>86.23(0.74)</td><td>61.91(0.76)</td><td>62.69(0.76)</td><td>67.74(1.21)</td><td>69.64(0.49) 68.72(0.58)</td></tr><tr><td rowspan="5"></td><td>RCimg1-7, p=0.5, λ=10 RCmix1-7,=10</td><td>81.15(0.76) 81.78(1.11)</td><td>59.56(0.79) 61.14(0.51)</td><td>62.42(0.59) 63.57(0.29)</td><td>71.74(0.43) 71.97(0.38)</td><td>69.62(0.24)</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Results below are not directly comparable due to different Deep-All implementations.</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Deep-All (our run)</td><td>88.40</td><td>66.26</td><td>66.58</td><td>59.40</td><td>70.16</td></tr><tr><td>PAR (our run)</td><td>88.40</td><td>65.19</td><td>68.58</td><td>61.86</td><td>71.10</td></tr><tr><td rowspan="2">Carlucci et al. (2019)</td><td>PAR (reported)</td><td>89.6</td><td>66.3</td><td>68.3</td><td>64.1</td><td>72.08</td></tr><tr><td>Deep-All</td><td>89.98</td><td>66.68</td><td>69.41</td><td>60.02</td><td>71.52</td></tr><tr><td rowspan="2">Li et al. (2018a)</td><td>Jigen Deep-All</td><td>89.00</td><td>67.63</td><td>71.71</td><td>65.18</td><td>73.38</td></tr><tr><td>MLDG (use domain labels)</td><td>86.67 88.00</td><td>64.91</td><td>64.28</td><td>53.08</td><td>67.24</td></tr><tr><td rowspan="2">Li et al. (2018c)</td><td></td><td></td><td>66.23</td><td>66.88</td><td>58.96</td><td>70.01</td></tr><tr><td>Deep-All CIDDG (use domain labels)</td><td>77.98 78.65</td><td>57.55 62.70</td><td>67.04 69.73</td><td>58.52 64.45</td><td>65.27 68.88</td></tr></table>
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+ Results. Tab. 2 shows significant improvements on Sketch for both RandConv variants. Sketch is the most challenging domain with no color and much less texture compared to the other 3 domains. The success on Sketch demonstrates that our methods can guide the DNN to learn global representations focusing on shapes that are robust to texture changes. Without using the consistency loss, $\mathrm { R C } _ { \mathrm { m i x 1 - 7 } }$ achieves the best overall result improving over Deep-All by ${ \sim } 4 \%$ but adding MultiAug does not further improve the performance. Adding the consistency loss with $\lambda = 1 0$ , $\mathrm { R C } _ { \mathrm { m i x 1 - 7 } }$ and $\mathrm { R C } _ { \mathrm { i m g 1 - 7 } , p = 0 . 5 }$ performs better on Sketch but degrades performance on the other 3 domains, so do GreyScale and ColorJitter. This observation will be discussed in Sec 4.4.
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+ 4.3 GENERALIZING AN IMAGENET MODEL TO IMAGENET-SKETCH Table 3: Accuracy of ImageNet-trained AlexNet on ImageNet-Sketch (IN-S) data. Our methods outperform PAR by $5 \%$ and are on par with a Stylized-ImageNet (SIN) trained model. Note that PAR was built on top of a stronger baseline than our model, and both PAR and SIN fine-tuned the baseline model which helped the performance, while we train RandConv model from scratch.
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+ <table><tr><td></td><td>Baseline (Wang et al.,2019a)</td><td>PAR (Wang et al.,2019a)</td><td>Baseline</td><td>RCimg1-7, p=0.5,=10</td><td>RCmix1-7, 入=10</td><td>SIN (Geirhos et al., 2019)</td></tr><tr><td>Top1</td><td>12.04</td><td>13.06</td><td>10.28</td><td>18.09</td><td>16.91</td><td>17.62</td></tr><tr><td>Top5</td><td>25.60</td><td>26.27</td><td>21.60</td><td>35.40</td><td>33.99</td><td>36.22</td></tr></table>
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+ ImageNet-Sketch (Wang et al., 2019a) is an out-of-domain test set for models trained on ImageNet. We trained AlexNet from scratch with $\mathrm { R C } _ { \mathrm { i m g 1 - 7 } , p = 0 . 5 , \lambda = 1 0 }$ and $\mathrm { R C } _ { \mathrm { m i x 1 - 7 } , \lambda = 1 0 }$ . We evaluate their performance on ImageNet-Sketch. We use the AlexNet model trained without RandConv as our baseline. Tab. 3 compares PAR and its baseline model and AlexNet trained with Stylized ImageNet (SIN) (Geirhos et al., 2019) on ImageNet-Sketch. Although PAR uses a stronger baseline, RandConv achieves significant improvements over our baseline and outperforms PAR by a large margin. Our methods achieve more than a $7 \%$ accuracy improvement over the baseline and surpass PAR by $5 \%$ . SIN as an image stylization approach that can modify image texture in a hierarchical and realistic way. However, albeit its complexity, it still performs on par with RandConv. Note that image stylization techniques require additional data and heavy precomputation. Further, the images for the style source also need to be chosen. In contrast, RandConv is much easier to use: it can be applied to any dataset via a simple convolution layer. We also measure the shape-bias metric proposed by Geirhos et al. (2019) for RandConv trained AlexNet. $\mathrm { R C } _ { \mathrm { i m g 1 - 7 } , p = 0 . 5 , \lambda = 1 0 }$ and $\mathsf { R C } _ { \operatorname* { m i x } 1 - 7 , \lambda = 1 0 }$ improve the baseline from $2 5 . 3 6 \%$ to $4 8 . 2 4 \%$ and $5 4 . 8 5 \%$ respectively.
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+ # 4.4 REVISITING PACS WITH MORE ROBUST PRETRAINED REPRESENTATIONS
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+ A common practice for many computer vision tasks (including the PACS benchmark) is transfer learning, i.e. finetuning a backbone model pretrained on ImageNet. Recently, how the accuracy on ImageNet (Kornblith et al., 2019) and adversial robustness (Salman et al., 2020) of the pretrained model affect transfer learning has been studied in the context of domain generalization. Instead, we study how out-of-domain generalizability transfers from pretraining to downstream tasks and shed light on how to better use pretrained models.
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+ Impact of ImageNet Pretraining A model trained on ImageNet may be biased towards textures (Geirhos et al., 2019). Finetuning ImageNet pretrained models on PACS may inherit this texture bias, thereby benefitting generalization on the Photo domain (which is similar to ImageNet), but hurting performance on the Sketch domain. Therefore, as shown in Sec. 4.2, using RandConv to correct this texture bias improves results on Sketch, but degrades them on the Photo domain. Since pretraining has such a strong impact on transfer performance to new tasks, we ask: "Can the generalizability of a pretrained model transfer to downstream tasks? I.e., does a pretrained model with better generalizability improve performance on unseen domains on new tasks?" To answer this, we revisit the PACS tasks based on ImageNet-pretrained weights where our two RandConv variants of Sec. 4.3 are used during ImageNet training. We study if this results in performance changes for the Deep-All baseline and for finetuning with RandConv.
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+ Better Performance via RandConv pretrained model We start by testing the Deep-All baselines using the two RandConv-trained ImageNet models of Sec. 4.3 as initialization. Tab. 4 shows significant improvements on Sketch. Results are comparable to finetuning with RandConv on a normal pretrained model. Art is also consistently improved. Performance drops slightly on Photo as expected, since we reduced the texture bias in the pretrained model, which is helpful for the Photo domain. A similar performance improvement is observed when using the SIN-trained AlexNet as initialization. Using RandConv for both ImageNet training and PACS finetuning, we achieve $7 6 . 1 1 \%$ accuracy on Sketch. As far as we know, this is the best performance using an AlexNet baseline. This approach even outperforms Jigen (Carlucci et al., 2019) $( 7 1 . 3 5 \% )$ with a stronger ResNet18 baseline model. Cartoon and Art are also improved. The best average domain generalization accuracy is $7 3 . 0 3 \%$ , with a more than $6 \%$ improvement over our initial Deep-All baseline.
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+ Table 4: Generalization results on PACS with RandConv and SIN pretrained AlexNet. ImageNet column shows how the pretrained model is trained on ImageNet (baseline represents training the ImageNet model using only the classification loss); PACS column indicates the methods used for finetuning on PACS. Best and second best accuracy for each target domain are highlighted in bold and underlined.
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+ <table><tr><td>PACS</td><td>ImageNet</td><td>Photo</td><td>Art</td><td>Cartoon</td><td>Sketch</td><td>Avg</td></tr><tr><td rowspan="4">Deep-All</td><td>Baseline</td><td>86.77(0.42)</td><td>60.11(1.33)</td><td>64.12(0.32)</td><td>55.28(4.71)</td><td>66.57(1.36)</td></tr><tr><td>RCimg1-7,p=0.5,=10</td><td>84.48(0.52)</td><td>62.61(1.23)</td><td>66.13(0.80)</td><td>69.24(0.80)</td><td>70.61(0.53)</td></tr><tr><td>RCmix1-7,λ=10</td><td>85.59(0.40)</td><td>63.30(0.99)</td><td>63.83(0.85)</td><td>68.29(1.27)</td><td>70.25(0.45)</td></tr><tr><td>SIN</td><td>85.33(0.66)</td><td>65.85(0.87)</td><td>65.39(0.62)</td><td>65.75(0.59)</td><td>70.58(0.21)</td></tr><tr><td rowspan="3">RCimg1-7, p=0.5,=10</td><td>Baseline</td><td>81.15(0.76)</td><td>59.56(0.79)</td><td>62.42(0.59)</td><td>71.74(0.43)</td><td>68.72(0.58)</td></tr><tr><td>RCimg1-7,p=0.5,入=10</td><td>84.36(0.36)</td><td>63.73(0.91)</td><td>68.07(0.55)</td><td>75.41(0.57)</td><td>72.89(0.33)</td></tr><tr><td>RCmix1-7,λ=10</td><td>84.63(0.97)</td><td>63.41(1.22)</td><td>66.36(0.43)</td><td>74.59(0.84)</td><td>72.25(0.54)</td></tr><tr><td rowspan="3">RCmix1-7 入=10</td><td>Baseline</td><td>81.78(1.11)</td><td>61.14(0.51)</td><td>63.57(0.29)</td><td>71.97(0.38)</td><td>69.62(0.24)</td></tr><tr><td>RCimg1-7,p=0.5,&gt;=10</td><td>85.16(1.03)</td><td>63.17(0.38)</td><td>67.68(0.60)</td><td>76.11(0.43)</td><td>73.03(0.46)</td></tr><tr><td>RCmix1-7,λ=10</td><td>86.17(0.56)</td><td>65.33(1.05)</td><td>65.52(1.13)</td><td>73.21(1.03)</td><td>72.56(0.50)</td></tr></table>
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+
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+ This experiment confirms that generalizability may transfer: removing texture bias may not only make a pretrained model more generalizable, but it may help generalization on downstream tasks. For similar target and pretraining domains like Photo and ImageNet, where learning texture bias may actually be beneficial, performance may degrade slightly.
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+ # 5 CONCLUSION AND DISCUSSION
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+ Randomized convolution (RandConv) is a simple but powerful data augmentation technique for randomizing local image texture. RandConv helps focus visual representations on global shape information rather than local texture. We theoretically justified the approximate shape-preserving property of RandConv and developed RandConv techniques using multi-scale and mixing designs. We also make use of a consistency loss to encourage texture invariance. RandConv outperforms state-of-the-art approaches on the digit recognition benchmark and on the sketch domain of PACS and on ImageNet-Sketch by a large margin. By finetuning a model pretrained with RandConv on PACS, we showed that the generalizability of a pretrained model may transfer to and benefit a new downstream task. This resulted in a new state-of-art performance on PACS in the Sketch domain.
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+ RandConv can help computer vision tasks when a shape-biased model is helpful e.g. for object detection. RandConv can also provide a shape-biased pretrained model to improve performance on downstream tasks when generalizing to unseen domains. However, local texture features can be useful for many computer vision tasks, especially for fixed-domain fine-grained visual recognition. In such cases, visual representations that are invariant to local texture may hurt in-domain performance. Therefore, important future work includes learning representations that disentangle shape and texture features and building models to use such representations in an explainable way.
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+ Adversarial robustness of deep neural networks has received significant recent attention. Interestingly, Zhang & Zhu (2019) find that adversarially-trained models are more shape biased; Shi et al. (2020) show that their method for increasing shape bias also helps adversarial robustness, especially when combined with adversarial training. Therefore, exploring how RandConv affects the adversarial robustness of models could be interesting future work. Moreover, recent biologically inspired models for improving adversarial robustness (Dapello et al., 2020) use Gabor filters with fixed random configurations followed by a stochastic layer to add Gaussian noise to the network input, which may explain the importance of randomness in RandConv. Exploring connections between RandConv and biological mechanisms in the human visual system would be interesting future work.
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+ Acknowledgments We thank Zhiding Yu for discussions on initial ideas and the experimental setup.
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+ We also thank Nathan Cahill for advice on proving the properties of random convolutions.
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+ ![](images/6c99ccc2a9d3fe69059bf93875355bdf5cbd0abb2fba9d4d9c3b1b8270762221.jpg)
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+ Figure 4: Left: An image with texture and shapes at different scales; Middle: The output of RandConv with a small filter size which largely preserves the shapes of the stones. Right: The output of RandConv with a large filter size distorts the shape of the stones as well.
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+ This supplementary material provides additional details. Specifically, in Sec. A and B, we discuss definitions of shapes and textures in images and justify why random convolution preserves global shapes and disrupts local texture formally by proving Theorem 1. This theorem shows that random linear projections are approximately distance preserving. We also discuss our simulation-based bound based on $80 \%$ distance rescaling on real image data. Sec. C provides more experimental details for the different datasets. Sec. D shows experimental results with a stronger backbone architecture and on a new benchmark ImageNet-R (Hendrycks et al., 2020a). Sec. E provides more detailed results regarding hyperparameter selection and ablation studies. Lastly, Sec. F shows example visualizations of RandConv outputs and for its mixing variant.
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+ # A SHAPES AND TEXTURE IN IMAGES
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+ As discussed in the main text, we define shapes in images that are preserved by a random convolution layer as primitive shapes: spatial clusters of pixels with similar local texture. An object in a image can be a single primitive shape alone but in most cases it is the composition of multiple primitive shapes e.g. a car includes wheels, body frames, windshields. Note that the definition of texture is not necessarily opposite to shapes, since the texture of a larger shape can includes smaller shapes. For example, in Fig.4, the left occluded triangle shape has texture composed by shapes of cobble stones while cobble stones have their own texture. Random convolution can preserve those large shapes that usually define the image semantics while distorting the small shapes as local texture.
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+ To formally define the shape-preserving property, we assume $( x _ { 1 } , y _ { 1 } )$ , $( x _ { 2 } , y _ { 2 } )$ and $( x _ { 3 } , y _ { 3 } )$ are three locations on a image and $( x _ { 1 } , y _ { 1 } )$ has closer color and local texture with $( x _ { 2 } , y _ { 2 } )$ than $( x _ { 3 } , y _ { 3 } )$ . For example, $( x _ { 1 } , y _ { 1 } )$ and $( x _ { 2 } , y _ { 2 } )$ are within the same shape while $( x _ { 3 } , y _ { 3 } )$ is located at a neighboring shape. Then we have kp $( x _ { 1 } , y _ { 1 } ) - \mathbf { p } ( x _ { 2 } , y _ { 2 } ) \| < \| \mathbf { p } ( x _ { 1 } , y _ { 1 } ) - \mathbf { p } ( x _ { 3 } , y _ { 3 } ) \|$ , where $\mathbf { p } ( x _ { i } , y _ { i } )$ is the image patch at location $( x _ { i } , y _ { i } )$ . A transformation $f$ is shape-preserving if it maintains such relative distance relations for most location triplets, i.e.
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+
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+ $$
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+ \| f ( \mathbf { p } ( x _ { i } , y _ { i } ) ) - f ( \mathbf { p } ( x _ { j } , y _ { j } ) ) \| / \| \mathbf { p } ( x _ { i } , y _ { i } ) - \mathbf { p } ( x _ { j } , y _ { j } ) \| \approx r
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+ $$
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+
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+ for any two spatial location $( x _ { i } , y _ { i } )$ and $( x _ { j } , y _ { j } ) ; r \ge 0$ is a constant.
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+
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+ # B RANDOM CONVOLUTION IS SHAPE-PRESERVING AS RANDOM LINEAR PROJECTION IS DISTANCE PRESERVING
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+
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+ We can express a convolution layer as a local linear projection:
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+
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+ $$
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+ \mathbf { g } ( x , y ) = \mathbf { U } \mathbf { p } ( x , y ) ,
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+ $$
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+
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+ where $ { \mathbf { p } } ( x , y ) \in { \mathbb { R } } ^ { d }$ ${ \bf \chi } d = h \times w \times C _ { i n } )$ is the vectorized image patch centerized at location $( x , y )$ , $\mathbf { g } ( x , y ) \in \mathbb { R } ^ { C _ { o u t } }$ is the output feature at location $( x , y )$ , and $\breve { \mathbf { U } } \in \breve { \mathbb { R } } ^ { C _ { o u t } \times d }$ is the matrix expressing the convolution layer filters $\Theta$ . I.e., for each sliding window centered at $( x , y )$ , a convolution layer applies a linear transform $f : \mathbb { R } ^ { d } \mathbb { R } ^ { C _ { o u t } }$ projecting the $d$ dimensional local image patch $\mathbf { p } ( x , y )$ to its $C _ { o u t }$ dimensional feature ${ \bf g } ( x , y )$ . When $\Theta$ is independently randomly sampled, e.g. from a Gaussian distribution, the convolution layer preserves global shapes since that a random linear projection is approximately distance-preserving by bounding the range of $r$ in Eq. 1 in Theorem 1.
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+
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+ Theorem 1. Suppose we have $N$ data points $\mathbf { z } _ { 1 } , \allowbreak \cdots , \allowbreak \mathbf { z } _ { N } \in \mathbb { R } ^ { d }$ . Let $f ( \mathbf { z } ) = \mathbf { U } \mathbf { z }$ be a random linear projection $f : \mathbb { R } ^ { d } \mathbb { R } ^ { m }$ such that $\mathbf { U } \in \mathbb { R } ^ { m \times d }$ and $\mathbf { U } _ { i , j } \sim N ( 0 , \sigma ^ { 2 } )$ . Then we have:
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+
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+ $$
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+ \begin{array} { r l } & { P \Big ( \underset { i \neq j ; i , j \in [ N ] } { \operatorname* { s u p } } \Big \{ r _ { i , j } : = \frac { \| f ( \mathbf { z } _ { i } ) - f ( \mathbf { z } _ { j } ) \| } { \| \mathbf { z } _ { i } - \mathbf { z } _ { j } \| } \Big \} > \delta _ { 1 } \Big ) \leq \epsilon , } \\ & { P \Big ( \underset { i \neq j ; i , j \in [ N ] } { \operatorname* { i n f } } \Big \{ r _ { i , j } : = \frac { \| f ( \mathbf { z } _ { i } ) - f ( \mathbf { z } _ { j } ) \| } { \| \mathbf { z } _ { i } - \mathbf { z } _ { j } \| } \Big \} < \delta _ { 2 } \Big ) \leq \epsilon , } \end{array}
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+ $$
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+
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+ where $\delta _ { 1 } : = \sigma \sqrt { \chi _ { \frac { 2 \epsilon } { N ( N - 1 ) } } ^ { 2 } ( m ) }$ and $\delta _ { 2 } : = \sigma \sqrt { \chi _ { 1 - \frac { 2 \epsilon } { N ( N - 1 ) } } ^ { 2 } \left( m \right) }$ . Here, $\chi _ { \alpha } ^ { 2 } ( m )$ denotes the $\alpha$ -upper quantile of the $\chi ^ { 2 }$ distribution with m degrees of freedom.
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+
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+ Thm. 1 tells us that for any data pair $( \mathbf { z } _ { i } , \mathbf { z } _ { j } )$ in a set of $N$ points, the distance rescaling ratio $r _ { i , j }$ after a random linear projection is bounded by $\delta _ { 1 }$ and $\delta _ { 2 }$ with probability $1 - \epsilon$ . A Smaller $N$ and a larger output dimension $m$ give better bounds. E.g., when $m = 3$ , $N = 1 , 0 0 0$ , $\sigma = 1$ and $\epsilon = 0 . 1$ , $\delta _ { 1 } = 5 . 8$ and $\delta _ { 2 } = 0 . 0 1$ . Thm. 1 gives a theoretical bound for all the $N ( N - 1 ) / 2$ pairs. However, in practice, preserving distances for a majority of $N ( N - 1 ) / 2$ pairs is sufficient. To empirically verify this, we test the range of central $8 0 \%$ of $\{ r _ { i , j } \}$ on real image data. Using the same $( m , N , \sigma , \epsilon )$ , $8 0 \%$ of the pairs lie in [0.56, 2.87], which is significantly better than the strict bound: [0.01, 5.8]. A proof of the theorem and simulation details are given in the following.
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+
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+ Proof. Let ${ \bf U } _ { k }$ represent to the $k$ -th row of $\mathbf { U }$ . It is easy to check that $\mathbf { v } _ { k } : = \langle \mathbf { U } _ { k } , \mathbf { z } _ { i } - \mathbf { z } _ { j } \rangle / \lVert \mathbf { z } _ { i } - \mathbf { z } _ { j } \rVert \sim$ $N ( 0 , \sigma ^ { 2 } )$ . Therefore,
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+
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+ $$
290
+ { \frac { \| f ( \mathbf { z } _ { i } ) - f ( \mathbf { z } _ { j } ) \| ^ { 2 } } { \sigma ^ { 2 } \| \mathbf { z } _ { i } - \mathbf { z } _ { j } \| ^ { 2 } } } = { \frac { 1 } { \sigma ^ { 2 } } } { \frac { ( \mathbf { z } _ { i } - \mathbf { z } _ { j } ) ^ { \top } \mathbf { U } ^ { \top } \mathbf { U } ( \mathbf { z } _ { i } - \mathbf { z } _ { j } ) } { \| \mathbf { z } _ { i } - \mathbf { z } _ { j } \| ^ { 2 } } } = \sum _ { k = 1 } ^ { m } { \frac { \mathbf { v } _ { k } ^ { 2 } } { \sigma ^ { 2 } } } \sim \chi ^ { 2 } ( m ) .
291
+ $$
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+
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+ Therefore, for $0 < \epsilon < 1$ , we have
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+
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+ $$
296
+ P \Big ( \frac { \| f ( \mathbf { z } _ { i } ) - f ( \mathbf { z } _ { j } ) \| ^ { 2 } } { \sigma ^ { 2 } \| \mathbf { z } _ { i } - \mathbf { z } _ { j } \| ^ { 2 } } > \chi _ { \frac { 2 \epsilon } { N ( N - 1 ) } } ^ { 2 } ( m ) \Big ) \leq \frac { 2 \epsilon } { N ( N - 1 ) } .
297
+ $$
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+
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+ From the above inequality, we have
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+
301
+ $$
302
+ \begin{array} { r l } & { P \Big ( \operatorname* { s u p } _ { i \not = j ; i , j \in [ N ] } \Big \{ \frac { \| f ( \alpha _ { i } ) - f ( z _ { j } ) \| ^ { 2 } } { \| z _ { i } - z _ { j } \| ^ { 2 } } \Big \} > \sigma ^ { 2 } \chi _ { \frac { 2 \epsilon } { N ( N - 1 ) } } ^ { 2 } ( m ) \Big ) } \\ & { = P \Big ( \operatorname* { s u p } _ { i \not = j ; i , j \in [ N ] } \Big \{ \frac { \| f ( \alpha _ { i } ) - f ( z _ { j } ) \| ^ { 2 } } { \sigma ^ { 2 } \| z _ { i } - z _ { j } \| ^ { 2 } } \Big \} > \chi _ { \frac { 2 \epsilon } { N ( N - 1 ) } } ^ { 2 } ( m ) \Big ) } \\ & { = P \Big ( \underset { i \not = j ; i , j \in [ N ] } { \bigcup } \Big \{ \frac { \| f ( \alpha _ { i } ) - f ( z _ { j } ) \| ^ { 2 } } { \sigma ^ { 2 } \| z _ { i } - z _ { j } \| ^ { 2 } } > \chi _ { \frac { 2 \epsilon } { N ( N - 1 ) } } ^ { 2 } ( m ) \Big \} \Big ) } \\ & { \le \underset { i \not = j ; i , j \in [ N ] } { \sum } P \Big ( \frac { \| f ( z _ { i } ) - f ( z _ { j } ) \| ^ { 2 } } { \sigma ^ { 2 } \| z _ { i } - z _ { j } \| ^ { 2 } } > \chi _ { \frac { 2 \epsilon } { N ( N - 1 ) } } ^ { 2 } ( m ) \Big ) } \\ & { \le \epsilon , } \end{array}
303
+ $$
304
+
305
+ which is equivalent to
306
+
307
+ $$
308
+ P \Big ( \operatorname* { s u p } _ { \substack { i \neq j ; i , j \in [ N ] } } \Big \{ \frac { \| f ( \mathbf { z } _ { i } ) - f ( \mathbf { z } _ { j } ) \| } { \| \mathbf { z } _ { i } - \mathbf { z } _ { j } \| } \Big \} > \sigma \sqrt { \chi _ { \frac { 2 \epsilon } { N ( N - 1 ) } } ^ { 2 } ( m ) } \Big ) \leq \epsilon .
309
+ $$
310
+
311
+ Similarly, we have
312
+
313
+ $$
314
+ P \Big ( \operatorname* { i n f } _ { \substack { i \neq j ; i , j \in [ N ] } } \Big \{ \frac { \| f ( \mathbf { z } _ { i } ) - f ( \mathbf { z } _ { j } ) \| } { \| \mathbf { z } _ { i } - \mathbf { z } _ { j } \| } \Big \} < \sigma \sqrt { \chi _ { 1 - \frac { 2 \epsilon } { N ( N - 1 ) } } ^ { 2 } ( m ) } \Big ) \leq \epsilon .
315
+ $$
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+
317
+ Simulation on Real Image Data To better understand the relative distance preservation property of random linear projections in practice, we use Algorithm 2 to empirically obtain a bound for real image data. We choose $m = 3$ , $N = 1 , 0 0 0$ , $\sigma = 1$ and $\epsilon = 0 . 1$ as in computing our theoretical bounds. We use $M = 1 , 0 0 0$ real images from the PACS dataset for this simulation. Note that the image patch size or $d$ does not affect the bound. We use a patch size of $3 \times 3$ resulting in $d = 2 7$ This simulation tell us that applying linear projections with a randomly sampled $U$ on $N$ local images patches in every image, we have a $1 - \epsilon$ chance that $8 0 \%$ of $r _ { i , j }$ is in the range $\left[ \delta _ { 1 0 \% } , \delta _ { 9 0 \% } \right]$ .
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+
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+ Algorithm 2 Simulate the range of central $80 \%$ of $\boldsymbol { r } _ { i , j }$ on real image data
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+
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+ 1: Input: $M$ images $\{ I _ { i } \} _ { i = 1 } ^ { M }$ , number of data points $N$ , projection output dimension $m$ , standard deviation $\sigma$ of normal distribution, confidence level $\epsilon$ .
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+ 2: for 3: $m = 1 M$ dos patches in at 1,000 locations and vectorize them as $I _ { m }$ $\{ \mathbf { z } _ { l } ^ { m } \} _ { l = 1 } ^ { N }$
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+ 4: Sample a projection matrix $\mathbf { U } \in \mathbb { R } ^ { m \times d }$ and $\mathbf { U } _ { i , j } \sim N ( 0 , \sigma ^ { 2 } )$
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+ 5: for $i = 1 N$ do
325
+ 6: for $j = i + 1 N$ do
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+ 7: Compute $\begin{array} { r } { r _ { i , j } ^ { m } = \frac { \| f ( \mathbf { z } _ { i } ^ { m } ) - f ( \mathbf { z } _ { j } ^ { m } ) \| } { \| \mathbf { z } _ { i } ^ { m } - \mathbf { z } _ { j } ^ { m } \| } } \end{array}$ , where $f ( \mathbf { z } ) = \mathbf { U } \mathbf { z }$
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+ 8: $q _ { 1 0 \% } ^ { m } = 1 0 \%$ quantile of $r _ { i , j } ^ { m }$ for $I _ { m }$
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+ 9: 10: $q _ { 9 0 \% } ^ { m } = 9 0 \%$ quantile of all $r _ { i , j } ^ { m }$ for $I _ { m }$ . Get the central $80 \%$ of $\boldsymbol { r } _ { i , j }$ in each image $\delta _ { 1 0 \% } = \epsilon$ $q _ { 1 0 \% } ^ { m }$
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+ 11: $\delta _ { 9 0 \% } = ( 1 - \epsilon )$ quantile of all $q _ { 9 0 \% } ^ { m }$ . Get the  confident bound for qm10% and qm90%
330
+ 12: return $\delta _ { 1 0 \% }$ , $\delta _ { 9 0 \% }$
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+
332
+ # C EXPERIMENTAL DETAILS
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+
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+ Digits Recognition The network for our digits recognition experiments is composed of two $C o n \nu { 5 } \times { 5 } .$ ReLU-MaxPool2 $^ { \prime } \times 2$ blocks with 64/128 output channels and three fully connected layer with 1024/1024/10 output channels. We train the network with batch size 32 for 10,000 iterations. During training, the model is validated every 250 iterations and saved with the best validation score for testing. We apply the Adam optimizer with an initial learning rate of 0.0001.
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+
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+ PACS We use the official data splits for training/validation/testing; no extra data augmentation is applied. We use the official PyTorch implementation and the pretrained weights of AlexNet for our PACS experiments. AlextNet is finetuned for 50,000 iterations with a batch size 128. Samples are randomly selected from the training data mixed between the three domains. We use the validation data of source domains only at every 100 iterations. We use the SGD optimizer for training with an initial learning rate of 0.001, Nesterov momentum, and weight decay set to 0.0005. We let the learning rate decay by a factor of 0.1 after finishing $80 \%$ of the iterations.
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+
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+ ImageNet Following the PyTorch example 3 on training ImageNet models, we set the batch size to 256 and train AlexNet from scratch for 90 epochs. We apply the SGD optimizer with an initial learning rate of 0.01, momentum 0.9, and weight decay 0.0001. We reduce the learning rate via a factor of 0.1 every 30 epochs.
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+
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+ # D MORE EXPERIMENTS WITH RESNET-18
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+
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+ In this section, we demonstrate that RandConv also works on other stronger backbone architectures, e.g. for a Residual Network He et al. (2016a). Specifically, we run the PACS and ImageNet experiments with ResNet-18 as the baseline and RandConv. As Table 5 shows, RandConv improves the baseline using ResNet18 on ImageNet-sketch by $1 0 . 5 \%$ accuracy. When using a RandConv pretrained ResNet-18 on PACS, the performance of finetuning with DeepAll and RandConv are both improved shown in Table 7. The best average domain generalization accuracy is $8 4 . 0 9 \%$ , with a more than $8 \%$ improvement over our initial Deep-All baseline. A model pretrained with $\mathrm { R C } _ { \mathrm { m i x 1 - 7 } , \lambda = 1 0 }$ generally performs better than when pretrained with $\mathrm { R C } _ { \mathrm { i m g 1 - 7 } , p = 0 . 5 , \lambda = 1 0 }$ . We also provide the ResNet-18 performance of JiGen (Carlucci et al., 2019) on PACS as reference. Note that JiGen uses extra data augmentation and a different data split than our approach and it only improves over its own baseline by $1 . 5 \%$ . In addition, we test RandConv trained ResNet-18 on ImageNet-R (Hendrycks et al., 2020a), a domain generalization benchmark that contains images of artistic renditions of 200 object classes from the original ImageNet dataset. As Table 6 shows, RandConv also improve the generalization performance on ImageNet-R and reduce the gap between the in-domain (ImageNet-200) and out-of-domain (ImageNet-R) performance.
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+
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+ Table 5: Accuracy of ImageNet-trained ResNet-18 on ImageNet-Sketch data.
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+
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+ <table><tr><td></td><td>Baseline</td><td>RCimg1-7,p=0.5, λ=10</td><td>RCmix1-7, 入=10</td></tr><tr><td>Top1</td><td>20.23</td><td>28.79</td><td>30.70</td></tr><tr><td>Top5</td><td>37.26</td><td>49.02</td><td>51.80</td></tr></table>
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+
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+ Table 6: Top 1 Accuracy of ImageNet-trained ResNet-18 on ImageNet-R data. ImageNet-200 are the original ImageNet data with the same 200 classes as ImageNet-R.
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+
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+ <table><tr><td></td><td>Baseline</td><td>RCimg1-7,p=0.5,&gt;=10</td><td>RCmix1-7, =10</td></tr><tr><td>ImageNet-200 (%)</td><td>88.15</td><td>83.72</td><td>72.7</td></tr><tr><td>ImageNet-R (%)</td><td>33.06</td><td>37.38</td><td>35.75</td></tr><tr><td>Gap</td><td>55.09</td><td>46.34</td><td>36.95</td></tr></table>
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+
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+ Table 7: Generalization results on PACS with RandConv pretrained model using ResNet-18. ImageNet column shows how the pretrained model is trained on ImageNet (baseline represents training using only the classification loss); PACS column indicates the methods used for finetuning on PACS. Best and second best accuracy for each target domain are highlighted in bold and underlined. The performance of JiGen (Carlucci et al., 2019) and its baseline using ResNet-18 is also given.
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+
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+ <table><tr><td>PACS</td><td>ImageNet</td><td>Photo</td><td>Art</td><td>Cartoon</td><td>Sketch</td><td>Avg</td></tr><tr><td rowspan="3">Deep-All</td><td>Baseline</td><td>95.45(0.43)</td><td>74.96(0.99)</td><td>71.48(1.22)</td><td>62.09(1.12)</td><td>76.00(0.37)</td></tr><tr><td>RCimg1-7,p=0.5,λ=10</td><td>94.65(0.16)</td><td>73.85(0.97)</td><td>74.78(0.58)</td><td>73.51(1.16)</td><td>79.20(0.40)</td></tr><tr><td>RCmix1-7,λ=10</td><td>94.10(0.43)</td><td>76.72(1.43)</td><td>73.41(1.29)</td><td>77.60(0.55)</td><td>80.46(0.74)</td></tr><tr><td rowspan="3">RCimg1-7, p=0.5,=10</td><td>Baseline</td><td>92.37(0.54)</td><td>76.50(0.55)</td><td>71.33(0.29)</td><td>79.65(1.32)</td><td>79.96(0.53)</td></tr><tr><td>RCimg1-7,p=0.5,=10</td><td>94.43(0.22)</td><td>79.80(1.03)</td><td>73.40(0.37)</td><td>81.51(0.85)</td><td>82.28(0.38)</td></tr><tr><td>RCmix1-7,λ=10</td><td>94.57(0.45)</td><td>81.32(1.00)</td><td>76.28(0.82)</td><td>84.18(0.94)</td><td>84.09(0.61)</td></tr><tr><td rowspan="3">RCmix1-7 入=10</td><td>Baseline</td><td>93.57(0.40)</td><td>77.73(0.91)</td><td>71.24(0.91)</td><td>75.53(2.17)</td><td>79.52(0.61)</td></tr><tr><td>RCimg1-7,p=0.5,入=10</td><td>95.23(0.30)</td><td>80.56(0.82)</td><td>74.18(0.53)</td><td>80.70(1.43)</td><td>82.67(0.46)</td></tr><tr><td>RCmix1-7,=10</td><td>95.01(0.32)</td><td>81.09(1.24)</td><td>76.04(0.92)</td><td>83.02(0.93)</td><td>83.79(0.60)</td></tr><tr><td rowspan="2">Deep-All JiGen</td><td rowspan="2">Baseline</td><td>95.73</td><td>77.85</td><td>74.86</td><td>67.74</td><td>79.05</td></tr><tr><td>96.03</td><td>79.42</td><td>75.25</td><td>71.35</td><td>80.51</td></tr></table>
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+
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+ # E HYPERPARAMETER SELECTIONS AND ABLATION STUDIES ON DIGITS RECOGNITION BENCHMARKS
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+
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+ We provide detailed experimental results for the digits recognition datasets. Table 8 shows results for different hyperameters $p$ for $\mathrm { R C _ { i m g 1 } }$ . Table 9 shows results for an ablation study on the multi-scale design for $\operatorname { R C } _ { \operatorname* { m i x } }$ and $\mathrm { R C } _ { \mathrm { i m g } , p = 0 . 5 }$ . Table 10 shows results for studying the consistency loss weight $\lambda$ for $\mathrm { R C } _ { \mathrm { m i x 1 - 7 } }$ and $\mathrm { R C } _ { \mathrm { i m g 1 - 7 } , p = 0 . 5 }$ . Tables 8, 9, and 10 correspond to Fig. 2 (a)(b)(c) in the main text respectively.
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+
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+ Table 8: Ablation study of hyperparameter $p$ for $\mathrm { R C _ { i m g 1 } }$ on digits recognition benchmarks. DG-Avg is the average performance on MNIST-M, SVHN, SYNTH and USPS. Best results are bold.
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+
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+ <table><tr><td></td><td>MNIST-10k</td><td>MNIST-M</td><td>SVHN</td><td>USPS</td><td>SYNTH</td><td>DG Avg</td><td>MNIST-C</td></tr><tr><td>Baseline</td><td>98.40(0.84)</td><td>58.87(3.73)</td><td>33.41(5.28)</td><td>79.27(2.70)</td><td>42.43(5.46)</td><td>53.50(4.23)</td><td>88.20(2.10)</td></tr><tr><td>RCimg1, p=0.9</td><td>98.68(0.06)</td><td>83.53(0.37)</td><td>53.67(1.54)</td><td>80.38(1.41)</td><td>59.19(0.85)</td><td>69.19(0.34)</td><td>89.79(0.44)</td></tr><tr><td>RCimg1,p=0.7</td><td>98.64(0.07)</td><td>84.17(0.61)</td><td>54.50(1.55)</td><td>80.85(0.91)</td><td>60.25(0.85)</td><td>69.94(0.50)</td><td>89.20(0.60)</td></tr><tr><td>RCimg1, p=0.5</td><td>98.72(0.08)</td><td>85.17(1.12)</td><td>55.97(0.54)</td><td>80.31(0.85)</td><td>61.07(0.47)</td><td>70.63(0.42)</td><td>88.66(0.62)</td></tr><tr><td>RCimg1, p=0.3</td><td>98.71(0.12)</td><td>85.45(0.87)</td><td>54.62(1.52)</td><td>79.78(1.40)</td><td>60.51(0.41)</td><td>70.09(0.60)</td><td>89.02(0.32)</td></tr><tr><td>RCimg1, p=0.1</td><td>98.66(0.06)</td><td>85.57(0.79)</td><td>54.34(1.52)</td><td>79.21(0.44)</td><td>60.18(0.63)</td><td>69.83(0.38)</td><td>88.53(0.38)</td></tr><tr><td>RCimg1,p=0</td><td>98.55(0.13)</td><td>86.27(0.42)</td><td>52.48(3.00)</td><td>79.01(1.11)</td><td>59.53(1.14)</td><td>69.32(1.19)</td><td>88.01(0.36)</td></tr></table>
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+
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+ Table 9: Ablation study of multi-scale RandConv on digits recognition benchmarks for $\mathrm { R C } _ { \mathrm { m i x } }$ and $\mathrm { R C } _ { \mathrm { i m g } , p = 0 . 5 }$ . Best entries for each variant are bold.
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+
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+ <table><tr><td></td><td>|MNIST-10k</td><td>MNIST-M</td><td>SVHN</td><td>USPS</td><td>SYNTH</td><td>DG Avg</td><td>MNIST-C</td></tr><tr><td>RCmix1</td><td>98.62(0.06)</td><td>83.98(0.98)</td><td>53.26(2.59)</td><td>80.57(1.09)</td><td>59.25(1.38)</td><td>69.26(1.35)</td><td>88.59(0.38)</td></tr><tr><td>RCmix1-3</td><td>98.76(0.02)</td><td>84.66(1.67)</td><td>55.89(0.83)</td><td>80.95(1.15)</td><td>60.07(1.05)</td><td>70.39(0.58)</td><td>89.80(0.94)</td></tr><tr><td>RCmix1-5</td><td>98.76(0.06)</td><td>84.32(0.43)</td><td>56.50(2.68)</td><td>81.85(1.05)</td><td>60.76(1.02)</td><td>70.86(0.86)</td><td>90.06(0.80)</td></tr><tr><td>RCmix1-7</td><td>98.82(0.06)</td><td>84.91(0.68)</td><td>55.61(2.63)</td><td>82.09(1.00)</td><td>62.15(1.30)</td><td>71.19(1.21)</td><td>90.30(0.44)</td></tr><tr><td>RCmix1-9</td><td>98.81(0.12)</td><td>85.13(0.72)</td><td>54.18(3.36)</td><td>82.07(1.28)</td><td>61.85(1.41)</td><td>70.81(1.24)</td><td>90.83(0.52)</td></tr><tr><td>RCimg1, p=0.5</td><td>98.66(0.05)</td><td>85.12(0.96)</td><td>55.59(0.29)</td><td>80.65(0.71)</td><td>60.85(0.48)</td><td>70.55(0.15)</td><td>89.00(0.45)</td></tr><tr><td>RCimg1-3,p=0.5</td><td>98.79(0.07)</td><td>85.36(1.04)</td><td>55.60(1.09)</td><td>80.99(0.99)</td><td>61.26(0.80)</td><td>70.80(0.86)</td><td>89.84(0.70)</td></tr><tr><td>RCimg1-5, p=0.5</td><td>98.83(0.07)</td><td>86.33(0.47)</td><td>54.99(2.48)</td><td>80.82(1.83)</td><td>62.61(0.75)</td><td>71.19(1.25)</td><td>90.70(0.43)</td></tr><tr><td>RCimg1-7, p=0.5</td><td>98.83(0.07)</td><td>86.08(0.27)</td><td>54.93(1.27)</td><td>81.58(0.74)</td><td>62.78(0.86)</td><td>71.34(0.61)</td><td>91.18(0.38)</td></tr><tr><td>RCimg1-9, p=0.5</td><td>98.80(0.12)</td><td>85.63(0.70)</td><td>52.82(2.01)</td><td>81.48(1.22)</td><td>62.55(0.74)</td><td>70.62(0.73)</td><td>90.79(0.48)</td></tr></table>
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+
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+ Table 10: Ablation study of consistency loss weight $\lambda$ on digits recognition benchmarks for $\mathrm { R C } _ { \mathrm { m i x 1 - 7 } }$ and $\mathrm { R C } _ { \mathrm { i m g 1 - 7 } , p = 0 . 5 }$ . DG-Avg is the average performance on MNIST-M, SVHN, SYNTH and USPS. Best results for each variant are bold.
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+
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+ <table><tr><td></td><td>入</td><td>MNIST-10k</td><td>MNIST-M</td><td>SVHN</td><td>USPS</td><td>SYNTH</td><td>DG Avg</td><td>MNIST-C</td></tr><tr><td rowspan="6">RCmix1-7</td><td>20</td><td>98.90 (0.05)</td><td>87.18 (0.81)</td><td>57.68 (1.64)</td><td>83.55 (0.83)</td><td>63.08 (0.50)</td><td>72.87 (0.47)</td><td>91.14 (0.53)</td></tr><tr><td>10</td><td>98.85 (0.04)</td><td>87.76 (0.83)</td><td>57.52 (2.09)</td><td>83.36 (0.96)</td><td>62.88 (0.78)</td><td>72.88 (0.58)</td><td>91.62 (0.77)</td></tr><tr><td>5</td><td>98.94 (0.09)</td><td>87.53 (0.51)</td><td>55.70 (2.22)</td><td>83.12 (1.08)</td><td>62.37 (0.98)</td><td>72.18 (1.04)</td><td>91.46 (0.50)</td></tr><tr><td>1</td><td>98.95 (0.05)</td><td>86.77 (0.79)</td><td>56.00 (2.39)</td><td>83.13 (0.71)</td><td>63.18 (0.97)</td><td>72.27 (0.82)</td><td>91.15 (0.42)</td></tr><tr><td>0.1</td><td>98.84 (0.07)</td><td>85.41 (1.02)</td><td>56.51 (1.58)</td><td>81.84 (1.14)</td><td>61.86 (1.44)</td><td>71.41 (0.98)</td><td>90.72 (0.60)</td></tr><tr><td>0</td><td>98.82 (0.06)</td><td>84.91 (0.68)</td><td>55.61 (2.63)</td><td>82.09 (1.00)</td><td>62.15 (1.30)</td><td>71.19 (1.21)</td><td>90.30 (0.44)</td></tr><tr><td rowspan="6">RCimg1-7,p=0.5</td><td>20</td><td>98.79 (0.04)</td><td>87.53 (0.79)</td><td>53.92 (1.59)</td><td>81.83 (0.70)</td><td>62.16 (0.37)</td><td>71.36 (0.49)</td><td>91.20 (0.53)</td></tr><tr><td>10</td><td>98.86 (0.05)</td><td>87.67 (0.37)</td><td>54.95 (1.90)</td><td>82.08 (1.46)</td><td>63.37 (1.58)</td><td>72.02 (1.15)</td><td>90.94 (0.51)</td></tr><tr><td></td><td>98.90 (0.04)</td><td>87.77 (0.72)</td><td>55.00 (1.40)</td><td>82.10 (0.55)</td><td>63.58 (1.33)</td><td>72.11 (0.62)</td><td>90.83 (0.71)</td></tr><tr><td>5</td><td>98.86 (0.04)</td><td>86.74 (0.32)</td><td>53.26 (2.99)</td><td>81.51 (0.48)</td><td>62.00 (1.15)</td><td>70.88 (0.93)</td><td>91.11 (0.62)</td></tr><tr><td>0.1</td><td>98.85 (0.14)</td><td>86.85 (0.31)</td><td>53.55 (3.63)</td><td>81.23 (1.02)</td><td>62.77 (0.80)</td><td>71.10 (1.31)</td><td>91.13 (0.69)</td></tr><tr><td>0</td><td>98.83 (0.07)</td><td>86.08 (0.27)</td><td>54.93 (1.27)</td><td>81.58 (0.74)</td><td>62.78 (0.86)</td><td>71.34 (0.61)</td><td>91.18 (0.38)</td></tr></table>
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+
372
+ # F MORE EXAMPLES OF RA N DCO N V DATA AUGMENTATION
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+
374
+ We provide additional examples of RandConv outputs for different convolution filter sizes in Fig. 6 and for its mixing variants at scale $k = 7$ with different mixing coefficients in Fig. 5. We observe that RandConv with different filter sizes retains shapes at different scales. The mixing strategy can continuously interpolate between the training domain and a randomly sampled domain.
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+
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+ ![](images/b51a4c5250677ed5324645f25d5c5926a998a7c82c5f58fd3a2d67246e4f246c.jpg)
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+ Figure 5: Examples of the RandConv mixing variant $\mathrm { R C } _ { \mathrm { m i x 7 } }$ on images of size $2 2 4 ^ { 2 }$ with different mixing coefficients $\alpha$ . When $\alpha = 1$ , the output is just the original image input;when $\alpha = 0$ , we use the output of the random convolution layer as the augmented image.
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+
379
+ $$
380
+ k = 1 \mathrm { ~ } k = 3 \mathrm { ~ } k = 5 \mathrm { ~ } k = 7 \mathrm { ~ } k = 1 1 \mathrm { ~ } k = 1 5
381
+ $$
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+
383
+ ![](images/613f26d3b91db3efa35f9b74d1d77d87aaa54b9c830838a0f79c1e5ed5f62ad2.jpg)
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+ Figure 6: RandConv data augmentation examples on images of size $2 2 4 ^ { 2 }$ . First column is the input image; following columns are convolution results using random filters of different sizes $k$ . We can see that the smaller filter sizes help maintain the finer shapes.
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+ # TOWARDS NEURAL NETWORKS THAT PROVABLY KNOW WHEN THEY DON’T KNOW
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+
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+ Alexander Meinke University of Tübingen
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+
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+ Matthias Hein University of Tübingen
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+
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+ # ABSTRACT
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+
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+ It has recently been shown that ReLU networks produce arbitrarily over-confident predictions far away from the training data. Thus, ReLU networks do not know when they don’t know. However, this is a highly important property in safety critical applications. In the context of out-of-distribution detection (OOD) there have been a number of proposals to mitigate this problem but none of them are able to make any mathematical guarantees. In this paper we propose a new approach to OOD which overcomes both problems. Our approach can be used with ReLU networks and provides provably low confidence predictions far away from the training data as well as the first certificates for low confidence predictions in a neighborhood of an out-distribution point. In the experiments we show that stateof-the-art methods fail in this worst-case setting whereas our model can guarantee its performance while retaining state-of-the-art OOD performance.1
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+
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+ # 1 INTRODUCTION
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+
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+ Deep Learning Models are being deployed in a growing number of applications. As these include more and more systems where safety is a concern, it is important to guarantee that deep learning models work as one expects them to. One topic that has received a lot of attention in this area is the problem of adversarial examples, in which a model’s prediction can be changed by introducing a small perturbation to an originally correctly classified sample. Achieving robustness against this type of perturbation is an active field of research. Empirically, adversarial training (Madry et al., 2018) performs well and provably robust models have been developed (Hein & Andriushchenko, 2017; Wong & Kolter, 2018; Raghunathan et al., 2018; Mirman et al., 2018; Cohen et al., 2019).
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+
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+ On the other end of the spectrum it is also important to study how deep learning models behave far away from the training samples. A simple property every classifier should satisfy is that far away from the training data, it should yield close to uniform confidence over the classes: it knows when it does not know. However, several cases of high confidence predictions far away from the training data have been reported for neural networks, e.g. fooling images (Nguyen et al., 2015), for out-of-distribution (OOD) images (Hendrycks & Gimpel, 2017a) or in medical diagnosis (Leibig et al., 2017). Moreover, it has been observed that, even on the original task, neural networks often produce overconfident predictions (Guo et al., 2017). Very recently, it has been shown theoretically that the class of ReLU networks (all neural networks which use a piecewise affine activation function), which encompasses almost all standard models, produces predictions with arbitrarily high confidences far away from the training data (Hein et al., 2019). Unfortunately, this statement holds for almost all such networks and thus without a change in the architecture one cannot avoid this phenomenon.
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+
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+ Traditionally, the calibration of the confidence of predictions has been considered on the indistribution (Guo et al., 2017; Lakshminarayanan et al., 2017a). However these techniques cannot be used for detection (Leibig et al., 2017). Only recently the detection of OOD inputs (Hendrycks & Gimpel, 2017a) has been tackled. The existing approaches are roughly of two types: first, postprocessing techniques that adjust the estimated confidence (DeVries & Taylor, 2018; Liang et al., 2018) which includes the baseline ODIN. Second, modification of the classifier training by integrating generative models like a VAE or GAN in order to discriminate out-distribution from in-distribution data (Lee et al., 2018a; Wang et al., 2018; Lee et al., 2018b) or approaches which enforce low confidence on OOD inputs during training (Hein et al., 2019; Hendrycks et al., 2019). Worst-case aspects of OOD detection have previously been studied in Nguyen et al. (2015); Schott et al. (2018); Hein et al. (2019); Sehwag et al. (2019), but no robustness guarantees have yet been proposed for this setting. A generalization guarantee for an out-of-distribution detection scheme is provided in Liu et al. (2018). While this is the only guarantee we are aware of, it is quite different from the type of guarantees we present in this paper. In particular, none of those approaches are able to guarantee that neural networks produce low confidence predictions far away from the training data. We prove that our classifier satisfies this requirement even when we use ReLU networks as the classifier model - without loosing performance on either the prediction task on the in-distribution nor the OOD detection performance, see Figure 1 for an illustration. Moreover, our technique allows to give upper bounds on the confidence over a whole neighborhood around a point (worst-case guarantees). We show that most state-of-the-art OOD methods can be fooled by maximizing the confidence in this ball even when starting from uniform noise images, which should be trivial to identify. The central difference from existing OOD-methods is that we have a Bayesian framework for in-and out-distribution, where we model in-and out-distribution separately. In this framework our algorithm for training neural networks follows directly as maximum likelihood estimator which is different from the more ad-hoc methods proposed in the literature. The usage of Gaussian mixture models as the density estimator is then the essential key to get the desired provable guarantees.
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+
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+ ![](images/747fe5b101b8bf494061e07d7c831cefdbbd333694847963a31a52ebc11f8fa6.jpg)
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+ Figure 1: Illustration on toy dataset: We show the color-coded confidence in the prediction (yellow indicates high confidence $\mathrm { m a x } _ { y } \hat { p } ( y | x ) \approx 1$ , whereas dark purple regions indicate low confidence $\operatorname* { m a x } _ { y } \hat { p } ( y | x ) \approx 0 . 5 )$ for a normal neural network (left) and our CCU neural network (right). The decision boundary is shown in white which is similar for both models. Our CCU-model retains high-confidence predictions in regions close to the training data, whereas far away from the training the CCU-model outputs close to uniform confidence. In contrast the normal neural network is over-confident everywhere except very close to the decision boundary.
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+
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+ # 2 A GENERIC MODEL FOR CLASSIFIERS WITH CERTIFIED LOW CONFIDENCE FAR AWAY FROM THE TRAINING DATA
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+
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+ The model which we propose in this paper assumes that samples from an out-distribution are given to us. In image recognition we could either see the set of all images as a sample from the outdistribution (Hendrycks et al., 2019) or consider the agnostic case where we use use uniform noise on $[ 0 , 1 ] ^ { d }$ as a maximally uninformative out-distribution. In both settings one tries to discriminate these out-distribution images from images coming from a particular image recognition task and the task is to get low confidence predictions on the out-distribution images vs. higher confidence on the images from the actual task. From the general model we derive under minimal assumptions a maximum-likelihood approach where one trains both a classifier for the actual task and density estimators for in- and out-distribution jointly. As all of these quantities are coupled in our model for the conditional distribution $p ( y | x )$ we get guarantees by controlling the density estimates far away from the training data. This is a crucial difference to the approaches of Lee et al. (2018a); Wang et al. (2018); Hendrycks et al. (2019) which empirically yield good OOD performance but are not able to certify the detection mechanism.
25
+
26
+ # 2.1 A PROBABILISTIC MODEL FOR IN- AND OUT-DISTRIBUTION DATA
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+
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+ We assume that there exists a joint probability distribution $p ( y , x )$ over the in- and out-distribution data, where $y$ are the labels in $\{ 1 , \dots , M \}$ , $M$ is the number of classes, and $x \in \mathbb { R } ^ { d }$ , where $d$ is the input dimension. In the following, we denote the underlying probabilities/densities with $p ( y | x )$ resp.
29
+
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+ $p ( x )$ and the estimated quantities with ${ \hat { p } } ( y | x )$ and ${ \hat { p } } ( x )$ . We are mainly interested in a discriminative framework, i.e. we want to estimate $p ( y | x )$ which one can represent via the conditional distribution of the in-distribution $p ( \boldsymbol { y } | \boldsymbol { x } , i )$ and out-distribution $p ( y | x , o )$ :
31
+
32
+ $$
33
+ p ( y | x ) = p ( y | x , i ) p ( i | x ) + p ( y | x , o ) p ( o | x ) = \frac { p ( y | x , i ) p ( x | i ) p ( i ) + p ( y | x , o ) p ( x | o ) p ( o ) } { p ( x | i ) p ( i ) + p ( x | o ) p ( o ) } .
34
+ $$
35
+
36
+ Note that at first it might seem strange to have a conditional distribution $p ( y | x , o )$ for out-distribution data, but until now we have made no assumptions about what in-and out-distribution are. A realistic scenario would be that at test time we are presented with instances $x$ from other classes (outdistribution) for which we expect a close to uniform $p ( y | x , o )$ .
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+
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+ Our model for ${ \hat { p } } ( y | x )$ has the same form as $p ( y | x )$
39
+
40
+ $$
41
+ \hat { p } ( y | x ) = \frac { \hat { p } ( y | x , i ) \hat { p } ( x | i ) \hat { p } ( i ) + \hat { p } ( y | x , o ) \hat { p } ( x | o ) \hat { p } ( o ) } { \hat { p } ( x | i ) \hat { p } ( i ) + \hat { p } ( x | o ) \hat { p } ( o ) } .
42
+ $$
43
+
44
+ Typically, out-distribution data has no relation to the actual task and thus we would like to have uniform confidence over the classes. Therefore we set in our model
45
+
46
+ $$
47
+ \hat { p } ( y | x , o ) = \frac { 1 } { M } \quad \mathrm { ~ a n d ~ } \quad \hat { p } ( y | x , i ) = \frac { e ^ { f _ { y } ( x ) } } { \sum _ { k = 1 } ^ { M } e ^ { f _ { k } ( x ) } } , \quad y \in \{ 1 , \dots M \} ,
48
+ $$
49
+
50
+ where $f : \mathbb { R } ^ { d } \to \mathbb { R } ^ { M }$ is the classifier function (logits). This framework is generic for classifiers trained with the cross-entropy (CE) loss (as the softmax function is the correct link function for the CE loss) and we focus in particular on neural networks. For a ReLU network the classifier function $f$ is componentwise a continuous piecewise affine function and has been shown to produce asymptotically arbitrarily highly confident predictions (Hein et al., 2019), i.e. the classifier gets more confident in its predictions the further it moves away from its training data. One of the main goals of our proposal is to fix this behavior of neural networks in a provable way.
51
+
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+ Note that with the choice of ${ \hat { p } } ( y | x , o )$ and non-zero priors for $\hat { p } ( i ) , \hat { p } ( o )$ , the full model ${ \hat { p } } ( y | x )$ can be seen as a calibrated version of ${ \hat { p } } ( y | x , i )$ , where $\hat { p } ( y | x ) \approx \hat { p } ( y | x , i )$ for inputs with ${ \hat { p } } ( x | i ) \gg { \hat { p } } ( x | o )$ and $\begin{array} { r } { \hat { p } ( y | x ) \approx \frac { 1 } { M } } \end{array}$ if $\hat { p } ( x | i ) \ll \hat { p } ( x | o )$ . However, note that only the confidence in the prediction ${ \hat { p } } ( y | x )$ is affected, the classifier decision is still done according to ${ \hat { p } } ( y | x , i )$ as the calibration does not change the ranking. Thus even if the OOD data came from the classification task we would like to solve, the trained classifier’s performance would be unaffected, only the confidence in the prediction would be damped.
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+
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+ For the marginal out-distribution ${ \hat { p } } ( x | o )$ there are two possible scenarios. In the first case one could concentrate on the worst case where we assume that $p ( x | o )$ is maximally uniformative (maximal entropy). This means that ${ \hat { p } } ( x | o )$ is uniform for bounded domains e.g. for images which are in $[ 0 , 1 ] ^ { \hat { d } }$ , ${ \hat { p } } ( x | o ) = 1$ for all $x \in [ 0 , 1 ] ^ { d }$ , or ${ \hat { p } } ( x | o )$ is a Gaussian for the domain of $\bar { \mathbb { R } } ^ { d }$ (the Gaussian has maximum entropy among all distributions of fixed variance). However, in this work we follow the approach of Hendrycks et al. (2019) where they used the 80 million tiny image dataset (Torralba et al., 2008) as a proxy of all possible images. Thus we estimate the density of ${ \hat { p } } ( x | o )$ using this data.
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+
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+ In order to get guarantees, the employed generative models for ${ \hat { p } } ( x | i )$ and ${ \hat { p } } ( x | o )$ have to be chosen in a way that allows one to control predictions far away from the training data. Variational autoencoders (VAEs) (Kingma & Welling, 2014; Rezende et al., 2014), normalizing flows (Dinh et al., 2016; Kingma & Dhariwal, 2018) and generative adversarial networks (GANs) (Goodfellow et al., 2014) are powerful generative models. However, there is no direct way to control the likelihood far away from the training data. Moreover, it has recently been discovered that VAEs, flows and GANs also suffer from overconfident likelihoods (Nalisnick et al., 2019; Hendrycks et al., 2019) far away from the data they are supposed to model as well as adversarial samples (Kos et al., 2017).
57
+
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+ For ${ \hat { p } } ( x | o )$ and ${ \hat { p } } ( x | i )$ we use a Gaussian mixture model (GMM) which is less powerful than a VAE but has the advantage that the density estimates can be controlled far away from the training data:
59
+
60
+ $$
61
+ \hat { p } ( x | i ) = \sum _ { k = 0 } ^ { K _ { i } } \alpha _ { k } \exp \left( - \frac { d ( x , \mu _ { k } ) ^ { 2 } } { 2 \sigma _ { k } ^ { 2 } } \right) , \qquad \hat { p } ( x | o ) = \sum _ { l = 0 } ^ { K _ { o } } \beta _ { l } \exp \left( - \frac { d ( x , \nu _ { l } ) ^ { 2 } } { 2 \theta _ { l } ^ { 2 } } \right)
62
+ $$
63
+
64
+ where $K _ { i } , K _ { o } \in \mathbb { N }$ are the number of centroids and $d : \mathbb { R } ^ { d } \times \mathbb { R } ^ { d } \mathbb { R }$ is the metric
65
+
66
+ $$
67
+ d ( x , y ) = \left. C ^ { - \frac { 1 } { 2 } } ( x - y ) \right. _ { 2 } ,
68
+ $$
69
+
70
+ with $C$ being a positive definite matrix and
71
+
72
+ $$
73
+ \alpha _ { k } = \frac { 1 } { K _ { i } } \frac { 1 } { ( 2 \pi \sigma _ { k } ^ { 2 } \operatorname * { d e t } C ) ^ { \frac { d } { 2 } } } , \quad \beta _ { l } = \frac { 1 } { K _ { o } } \frac { 1 } { ( 2 \pi \theta _ { l } ^ { 2 } \operatorname * { d e t } C ) ^ { \frac { d } { 2 } } } .
74
+ $$
75
+
76
+ We later fix $C$ as a slightly modified covariance matrix of the in-distribution data (see Section 4 for details). Thus one just has to estimate the centroids $\mu _ { k } , \nu _ { l }$ and the variances $\sigma _ { k } ^ { 2 } , \theta _ { l } ^ { 2 }$ . The idea of this metric is to use distances adapted to the data-distribution. Note that equation 4 is a properly normalized density in $\mathbb { R } ^ { d }$ .
77
+
78
+ # 2.2 MAXIMUM LIKELIHOOD ESTIMATION
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+
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+ Given models for ${ \hat { p } } ( y | x )$ and ${ \hat { p } } ( x )$ we effectively have a full generative model and apply maximum likelihood estimation to get the underlying classifier ${ \hat { p } } ( y | x , i )$ and the parameters of the Gaussian mixture models $\hat { p } ( x | i ) , \hat { p } ( x | o )$ . The only free parameter left is the probability $\hat { p } ( i ) , \hat { p } ( o )$ which we write compactly as $\begin{array} { r } { \lambda = \frac { \hat { p } ( o ) } { \hat { p } ( i ) } } \end{array}$ . In principle this parameter should be set considering the potential cost of over-confident predictions. In our experiments we simply fix it to $\lambda = 1$ .
81
+
82
+ $$
83
+ \begin{array} { r l } & { \qquad \underset { ( x , y ) \sim p ( x , y ) } { \mathbb { E } } \log \Big ( \hat { p } ( y , x ) \Big ) = \underset { ( x , y ) \sim p ( x , y ) } { \mathbb { E } } \log \big ( \hat { p } ( y | x ) \big ) + \log ( \hat { p } ( x ) ) , } \\ & { \qquad = \underset { ( x , y ) \sim p ( x , y ) } { \mathbb { E } } \log \Big ( \frac { \hat { p } ( y | x , i ) \hat { p } ( x | i ) \hat { p } ( i ) + \frac { 1 } { M } \hat { p } ( x | o ) \hat { p } ( o ) } { \hat { p } ( x | i ) \hat { p } ( i ) + \hat { p } ( x | o ) \hat { p } ( o ) } \Big ) + \log \big ( \hat { p } ( x | i ) \hat { p } ( i ) + \hat { p } ( x | o ) \hat { p } ( o ) \big ) . } \end{array}
84
+ $$
85
+
86
+ In practice, we have to compute empirical expectations from finite training data from the indistribution $( x _ { i } , y _ { i } ) _ { i = 1 } ^ { n _ { i } }$ and out-distribution $( z _ { j } ) _ { j = 1 } ^ { \bar { n } _ { o } }$ . Labels for the out-distribution could be generated randomly via $\textstyle p ( y | x , o ) = { \frac { 1 } { M } }$ , but we obtain an unbiased estimator with lower variance by averaging over all classes directly, as was done in Lee et al. (2018a); Hein et al. (2019); Hendrycks et al. (2019). Now we can estimate the classifier $f$ and the mixture model parameters $\mu , \nu , \sigma , \theta$ via
87
+
88
+ $$
89
+ \begin{array} { c l } { \displaystyle \underset { f , \mu , \nu , \sigma , \theta } { \arg \operatorname* { m a x } } \left\{ \frac { 1 } { n _ { i } } \sum _ { i = 1 } ^ { n _ { i } } \log \left( \hat { p } ( y _ { i } | x _ { i } ) \right) + \frac { \lambda } { n _ { o } } \sum _ { j = 1 } ^ { n _ { o } } \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \log \left( \hat { p } ( m | z _ { j } ) \right) \right. } \\ { \displaystyle \left. + \frac { 1 } { n _ { i } } \sum _ { i = 1 } ^ { n _ { i } } \log ( \hat { p } ( x _ { i } ) ) + \frac { \lambda } { n _ { o } } \sum _ { j = 1 } ^ { n _ { o } } \log ( \hat { p } ( z _ { j } ) ) \right\} , } \end{array}
90
+ $$
91
+
92
+ with
93
+
94
+ $$
95
+ \hat { p } ( y | x ) = \frac { \hat { p } ( y | x , i ) \hat { p } ( x | i ) + \frac { \lambda } { M } \hat { p } ( x | o ) } { \hat { p } ( x | i ) + \lambda \hat { p } ( x | o ) } \quad \mathrm { ~ a n d ~ } \quad \hat { p } ( x ) = \frac { 1 } { \lambda + 1 } \Big ( \hat { p } ( x | i ) + \lambda \hat { p } ( x | o ) \Big ) .
96
+ $$
97
+
98
+ Due to the bounds derived in Section 3, we denote our method by Certified Certain Uncertainty (CCU). Note that if one uses a standard neural network model with softmax, i.e. $\begin{array} { r } { \hat { p } ( y | x ) = \hat { p } ( y | x , i ) = \frac { e ^ { f _ { y } ( x ) } } { \sum _ { m = 1 } ^ { M } e ^ { f _ { m } ( x ) } } } \end{array}$ efy(x)PMm=1 efm(x) , then the first term in equation 6 would be the cross-entropy loss for the in-distribution data and the second term the cross entropy loss for the out-distribution data with a uniform distribution over the classes. For this choice of ${ \hat { p } } ( y | x )$ and neglecting the terms for ${ \hat { p } } ( x )$ we recover the approach of Hein et al. (2019); Hendrycks et al. (2019) for training a classifier which outputs uniform confidence predictions on out-distribution data where $\begin{array} { r } { \frac { \hat { p } ( i ) } { \hat { p } ( o ) } } \end{array}$ corresponds to that regularization parameter $\lambda$ . The key difference in our approach is that $\hat { p } ( y | x ) \neq \hat { p } ( y | x , i )$ and the estimated densities for in- and out distribution ${ \hat { p } } ( x | i )$ and ${ \hat { p } } ( x | o )$ lead to a confidence calibration of ${ \hat { p } } ( y | x )$ , and in turn the fit of the classifier influences the estimation of ${ \hat { p } } ( x | i )$ and ${ \hat { p } } ( x | o )$ . The major advantage of our model is that we can give guarantees on the confidence of the classifier decision far away from the training data.
99
+
100
+ # 3 PROVABLE GUARANTEES FOR CLOSE TO UNIFORM PREDICTIONS FAR AWAY FROM THE TRAINING DATA
101
+
102
+ In this section we provide two types of guarantees on the confidence of a classifier trained according to our model in equation 6. The first one says that the classifier has provably low confidence far away from the training data, where an explicit bound on the minimal distance is provided, and the second provides an upper bound on the confidence in a ball around a given input point. The latter bound resembles robustness guarantees for adversarial samples (Hein & Andriushchenko, 2017; Wong & Kolter, 2018; Raghunathan et al., 2018; Mirman et al., 2018) and is quite different from the purely empirical evaluation done in OOD detection papers as we show in Section 4.
103
+
104
+ We provide our bounds for a more general mixture model which includes our GMM in equation 4 as a special case. To our knowledge, these are the first such bounds for neural networks and thus it is the first modification of a ReLU neural network so that it provably “knows when it does not know” (Hein et al., 2019) in the sense that far away from the training data the predictions are close to uniform over the classes.
105
+
106
+ Theorem 3.1. Let $( x _ { i } ^ { ( i ) } , y _ { i } ^ { ( i ) } ) _ { i = 1 } ^ { n }$ be the training set of the in-distribution and let the model for the conditional probability be given as
107
+
108
+ $$
109
+ \forall x \in \mathbb { R } ^ { d } , y \in \{ 1 , \dots , M \} , \qquad { \hat { p } } ( y | x ) = { \frac { { \hat { p } } ( y | x , i ) { \hat { p } } ( x | i ) + { \frac { \lambda } { M } } { \hat { p } } ( x | o ) } { { \hat { p } } ( x | i ) + \lambda { \hat { p } } ( x | o ) } } ,
110
+ $$
111
+
112
+ where $\begin{array} { r } { \lambda = \frac { \hat { p } ( o ) } { \hat { p } ( i ) } > 0 } \end{array}$ and let the model for the marginal density of the in-distribution ${ \hat { p } } ( x | i )$ and out-distribution $p ( x | o )$ be given by the generalized GMMs
113
+
114
+ $$
115
+ \hat { p } ( x | i ) = \sum _ { k = 0 } ^ { K _ { i } } \alpha _ { k } \exp \left( - \frac { d ( x , \mu _ { k } ) ^ { 2 } } { 2 \sigma _ { k } ^ { 2 } } \right) , \qquad \hat { p } ( x | o ) = \sum _ { l = 0 } ^ { K _ { o } } \beta _ { l } \exp \left( - \frac { d ( x , \nu _ { l } ) ^ { 2 } } { 2 \theta _ { l } ^ { 2 } } \right)
116
+ $$
117
+
118
+ with $\alpha _ { k } , \beta _ { l } ~ > ~ 0$ and $\mu _ { k } , \nu _ { l } \ \in \ \mathbb { R } ^ { d } \quad \forall k \ = \ 1 , \dots K _ { i }$ , $l = 1 , \ldots , K _ { o }$ and $d : \mathbb { R } ^ { d } \times \mathbb { R } ^ { d } $ $\mathbb { R } _ { + }$ a metric. Let $z ~ \in ~ \mathbb { R } ^ { d }$ and define k∗ = arg min d(z,µk)σ , i∗ = arg min d(z, xi), $\begin{array} { r } { l ^ { * } = \underset { l = 1 , \ldots , K _ { o } } { \arg \operatorname* { m a x } } \beta _ { l } \exp \left( - \frac { d ( z , \nu _ { l } ) ^ { 2 } } { 2 \theta _ { l } ^ { 2 } } \right) } \end{array}$ and $\begin{array} { r } { \Delta = \frac { \theta _ { l ^ { * } } ^ { 2 } } { \sigma _ { k ^ { * } } ^ { 2 } } - 1 } \end{array}$ . For any $\epsilon > 0$ , $i f \operatorname* { m i n } _ { l } \theta _ { l } > \operatorname* { m a x } _ { k } \sigma _ { k }$ and
119
+
120
+ $$
121
+ \operatorname* { m i n } _ { i = 1 , \ldots , n } d ( z , x _ { i } ) \geq d ( x _ { i } * , \mu _ { k ^ { * } } ) + d ( \mu _ { k ^ { * } } , \nu _ { l ^ { * } } ) \Big [ \frac { 2 } { \Delta } + \frac { 1 } { \sqrt { \Delta } } \Big ] + \theta _ { l ^ { * } } \sqrt { \frac { 2 } { \Delta } \log \Big ( \frac { M - 1 } { \epsilon \lambda } \frac { \sum _ { k } \alpha _ { k } } { \beta _ { l ^ { * } } } \Big ) } ,
122
+ $$
123
+
124
+ then it holds for all $m \in \{ 1 , \ldots , M \}$ that
125
+
126
+ $$
127
+ \hat { p } ( m | z ) \leq \frac { 1 } { M } \big ( 1 + \epsilon \big ) .
128
+ $$
129
+
130
+ In particular, $i f \operatorname* { m i n } _ { i } d ( z , x _ { i } ) \to \infty$ , then $\begin{array} { r } { \hat { p } ( m | z ) \to \frac { 1 } { M } } \end{array}$ .
131
+
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+ The proof is given in Appendix A. Theorem 3.1 holds for any multi-class classifier which defines for each input a probability distribution over the labels. Given the parameters of the GMM’s it quantifies at which distance of an input $z$ to the training set the classifier achieves close to uniform confidence. The theorem holds even if we use ReLU classifiers which in their unmodified form have been shown to produce arbitrarily high confidence far away from the training data Hein et al. (2019). This is a first step towards neural networks which provably know when they don’t know.
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+ In the next corollary, we provide an upper bound on the confidence over a ball around a given data point. This allows to give “confidence guarantees” for a whole volume and thus is much stronger than the usual pointwise evaluation of OOD methods.
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+ Corollary 3.1. Let $x _ { 0 } \in \mathbb { R } ^ { d }$ and $R > 0$ , then with $\begin{array} { r } { \lambda = \frac { \hat { p } ( o ) } { \hat { p } ( i ) } } \end{array}$ it holds
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+
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+ $$
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+ \begin{array} { r l } & { \underset { d ( x , x _ { 0 } ) \leq R } { \operatorname* { m a x } } \ \hat { p } ( y | x ) \leq \frac { 1 } { M } \frac { 1 + M \frac { b } { \lambda } } { 1 + \frac { b } { \lambda } } , } \\ & { b = \frac { \sum _ { k = 1 } ^ { K _ { i } } \alpha _ { k } \exp { \left( - \frac { \operatorname* { m a x } \left\{ d ( x _ { 0 } , \mu _ { k } ) - R , 0 \right\} ^ { 2 } } { 2 \sigma _ { k } ^ { 2 } } \right) } } { \sum _ { l = 1 } ^ { K _ { o } } \beta _ { l } \exp { \left( - \frac { \left( d ( x _ { 0 } , \nu _ { l } ) + R \right) ^ { 2 } } { 2 \theta _ { l } ^ { 2 } } \right) } } . } \end{array}
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+ $$
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+
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+ The proof is in the Appendix B. We show in Section 4 that even though OOD methods achieve low confidence on noise images, the maximization of the confidence in a ball around a noise point (adversarial noise) yields high confidence predictions for OOD methods, whereas our classifier has provably low confidence, as certified by Corollary 3.1. The failure of OOD methods shows that the certification of entire regions is an important contribution of CCU which goes beyond the purely sampling-based evaluation.
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+ ![](images/a5fe0f60b215d258753bd6354e157b4507108b72b5077f8ee686d357b36284e2.jpg)
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+ Figure 2: Adversarial Noise: We maximize the confidence of the OOD methods using PGD in the ball around a uniform noise sample (seed images, left) on which CCU is guaranteed by Corollary 3.1 to yield less than $1 . 1 \frac { 1 } { M }$ maximal confidence. For each OOD method we report the image with the highest confidence. Maha and MCD use scores where lower is more confident (indicated by $^ *$ ). If we do not find a sample that has higher confidence/lower score than the median of the in-distribution, we highlight this in boldface. All other OOD methods fail on some dataset, see Table 1 for a quantitative version. ODIN at high temperatures always returns low confidence, so a value of 0.1 is not informative.
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+
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+ # 4 EXPERIMENTS
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+ We evaluate the worst-case performance of various OOD detection methods within regions for which CCU yields guarantees and by standard OOD on MNIST (LeCun et al., 1998), FashionMNIST (Xiao et al., 2017), SVHN (Netzer et al., 2011), CIFAR10 and CIFAR100 (Krizhevsky & Hinton, 2009). We show that all other OOD methods yield undesired high confidence predictions in the certified low confidence regions of CCU and thus would not detect these inputs as out-distribution. For calibrating hyperparameters resp. training we use for all OOD methods the 80 Million Tiny Images (Torralba et al., 2008) as out-distribution Hendrycks et al. (2019) which yields a fair and realistic comparison.
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+ CCU: As the Euclidean metric is known to be a relatively bad distance between two images we instead use the distance $d ( x , y ) = \left\| C ^ { - \frac { 1 } { 2 } } ( x - y ) \right\|$ , where $C$ is generated as follows. We calculate the covariance matrix $C ^ { \prime }$ on augmented in-distribution samples (see C.1). Let $( \lambda _ { i } , u _ { i } ) _ { i = 1 } ^ { d }$ be the eigenvalues/eigenvectors of $C ^ { \prime }$ . Then we set
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+
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+ $$
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+ C = \sum _ { i = 1 } ^ { d } \operatorname* { m a x } \{ \lambda _ { i } , 1 0 ^ { - 6 } \operatorname* { m a x } _ { j } \lambda _ { j } \} u _ { i } u _ { i } ^ { T } ,
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+ $$
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+
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+ that is we fix a lower bound on the smallest eigenvalue so that $C$ has full rank. In Hendrycks & Gimpel (2017b) a similar metric has been used for detection of adversarial images. We choose $K _ { i } =$ $K _ { o } = 1 0 0$ as the number of centroids for the GMMs. We initialize the in-GMM on augmented in-data using the EM algorithm with spherical covariance matrices in the transformed space, as in equation 4. For the out-distribution we use a subset of 20000 points for the initialization. While, initially it holds that $\forall k , l : \sigma _ { k } < \theta _ { l }$ , as required in Theorem 3.1, this is not guaranteed during the optimization of equation 6. Thus, we enforce the constraint during training by setting: $\theta _ { l } \mapsto \mathrm { m a x } \{ \theta _ { l } , \bar { 2 } \mathrm { m a x } _ { k } \sigma _ { k } \}$ at every gradient step. Since the “classifier” and “density” terms in equation 6 have very different magnitudes we choose a small learning rate of $1 e - 5$ for the parameters in the GMMs. It is also crucial to not apply weight decay to these parameters. The other hyperparameters are chosen as in the base model below.
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+ Benchmarks: For all OOD methods we use LeNet on MNIST and a Resnet18 (for GAN and MCD we use VGG) otherwise. The hyperparameters used during training can be found in Appendix
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+ Table 1: Worst-case performance of different OOD methods in neighborhoods around uniform noise points certified by CCU. We report the clean test error (TE) on the in-distribution (GAN and MCD use VGG). The success rate (SR) is the fraction of adversarial noise points for which the confidence/score inside the ball is higher than the median of the in-distribution’s confidence/score. The AUC quantifies detection of adversarial noise versus in-distribution. All values in $\%$ .
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+ <table><tr><td></td><td>Base</td><td>MCD</td><td>EDL 0.4</td><td>DE 0.4</td><td>GAN 0.8</td><td>ODIN 0.5</td><td>Maha 0.9</td><td>ACET</td><td>OE 0.7</td><td>CCU</td></tr><tr><td>LSINN TE SR</td><td>AUC</td><td>0.5 0.4 100.0 99.0 1.4 8.6 5.8</td><td>100.0 0.0 5.2</td><td>100.0 7.3 4.9</td><td>43.5 54.4 5.7</td><td>100.0 0.0</td><td>100.0 11.7</td><td>0.6 0.0 100.0</td><td>100.0 35.2</td><td>0.6 0.0 100.0</td></tr><tr><td>LSINII</td><td>TE SR AUC TE</td><td>4.8 100.0 0.0 2.9</td><td>72.5 100.0 47.1 0.0 3.9 3.1</td><td>100.0 0.4</td><td>99.0 39.5 4.2</td><td>4.8 100.0 0.0 2.9</td><td>4.8 100.0 18.8 2.9</td><td>4.8 0.0 100.0 3.2</td><td>5.7 100.0 35.7 4.1</td><td>4.9 0.0 100.0</td></tr><tr><td>NH∧S SR</td><td>AUC</td><td>100.0 73.5 0.0 34.1</td><td>100.0 0.0</td><td>2.4 100.0 0.0</td><td>0.0 100.0</td><td>100.0 0.0</td><td>100.0 0.0</td><td>3.0 96.5</td><td>100.0 0.0</td><td>3.0 0.0 100.0</td></tr><tr><td>CTTIIIIEITITIT</td><td>TE SR AUC</td><td>5.6 11.7 100.0 90.5 0.0 23.9</td><td>7.0 100.0 0.0</td><td>6.7 100.0 0.0</td><td>11.7 100.0 25.3</td><td>5.6 100.0 0.0</td><td>5.6 100.0 0.0</td><td>6.1 0.0 99.9</td><td>4.7 100.0 0.0</td><td>5.8 0.0 100.0</td></tr><tr><td></td><td>TE SR AUC</td><td>23.3 45.3 100.0 100.0 0.1</td><td>31.1 100.0 17.3 0.0</td><td>27.5 100.0 0.2</td><td>43.8 89.5 15.3</td><td>23.3 100.0 0.0</td><td>23.2 100.0 0.0</td><td>25.2 3.5 95.8</td><td>24.7 100.0 2.5</td><td>25.9 0.0 100.0</td></tr></table>
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+ C. The AUC (area under ROC) is computed by treating in-distribution versus out-distribution as a two-class problem using the confidence/score of the method as criterion. Alternatively one could report the AUPR (area under precision-recall curve) which we do in Appendix G. MCD: MonteCarlo Dropout (Gal & Ghahramani, 2016) uses dropout at train and at test time. Since it is not clear where to put the dropout layers in a ResNet, we use VGG instead. We take the softmax from 7 forward passes (Shafaei et al., 2018) and use the mean of the output for prediction and the variance as score. EDL: Evidential deep learning (Sensoy et al., 2018) replaces the softmax layer of a neural network and introduces a different loss function that encourages better uncertainty estimates. $\pmb { D } \pmb { E }$ : Deep ensembles (Lakshminarayanan et al., 2017b) average the softmax outputs of five models that were adversarially trained via FGSM (Goodfellow et al., 2015) with step size $\epsilon = 0 . 0 1$ . $\mathbf { G } A N .$ : The framework of confidence-calibrated classifiers (Lee et al., 2017) relies on training a GAN alongside a classifier such that the GAN’s generator is encouraged to generate points close to but not on the in-distribution. On these points one then enforces uniform confidence. We used their provided code to train a VGG this way, as we were unable to adapt the method to a ResNet with an acceptable test error (e.g. $\mathrm { T E } < 3 0 \%$ on SVHN). ODIN: ODIN (Liang et al., 2017) consists of two parts: a temperature $T$ by which one rescales the logits before the softmax layer P efk/T and a preprocessing step that applies a single FGSM-step (Goodfellow et al., 2015) of length $\epsilon$ before evaluating the input. The two parameters are calibrated on the out-distribution. Maha: The approach in Lee et al. (2018c) is based on computing a class-conditional Mahalanobis distance in feature space and applying an ODIN-like preprocessing step for each layer. Following Ren et al. (2019) we use a single-layer version of Lee et al. (2018c) on our networks’ penultimate layers because the multi-layer version in the original code does not support gradient-based attacks. $o E$ : Outlier exposure (Hendrycks et al., 2019) enforces uniform confidence on a large out-distribution. We use their provided code to train a model with our chosen architecture. ACET: Adversarial confidence enhanced training (ACET) (Hein et al., 2019) enforces low confidence on a ball around points from an out-distribution by running adversarial attacks during training. In order to make the comparison with OE more meaningful we use 80M tiny images to draw the seeds rather than smoothed uniform noise as in Hein et al. (2019). We refer to Appendix F for a discussion of the influence of this choice on the results.
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+ Some of the above OOD papers optimize their hyperparameters on a validation set for each outdistribution they test on. However, this leads to different classifiers for each out-distribution dataset which seems unrealistic as we want to have good generic OOD performance and not for a particular dataset. Thus we keep the comparison realistic and fair by calibrating the hyperparameters of all methods on a subset of 80M tiny images and then evaluating on the other unseen distributions.
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+ Table 2: AUC (in- versus out-distribution detection based on confidence/score) in percent for different OOD methods and datasets (higher is better). OE and CCU have the best OOD performance.
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+ <table><tr><td></td><td></td><td>Base MCD</td><td>EDL</td><td>DE</td><td>GAN</td><td>ODIN</td><td>Maha</td><td>ACET</td><td>OE</td><td>CCU</td></tr><tr><td>LSINN</td><td>FMNIST EMNIST GrCIFAR10 Noise Uniform</td><td>97.4 89.2 99.7 100.0 95.2</td><td>93.1 82.0 94.7 95.2 87.9</td><td>99.3 89.0 99.7 99.9 99.9</td><td>99.2 92.1 100.0 100.0 97.9</td><td>99.4 92.8 99.1 99.3 99.9</td><td>98.7 88.9 99.9 100.0 98.2</td><td>96.8 91.6 98.7 97.2 100.0</td><td>100.0 95.0 100.0 100.0 100.0 100.0 100.0 100.0</td><td>99.9 99.9 95.8 92.0 100.0 100.0 100.0</td></tr><tr><td>JSINII</td><td>MNIST EMNIST GrCIFAR10 Noise Uniform</td><td>96.7 97.5 91.0 97.3 96.9</td><td>82.7 87.3 92.3 94.0 93.3 91.9</td><td>94.5 95.6 84.0 95.6 95.6</td><td>96.7 97.1 86.1 97.4 98.3</td><td>99.9 99.9 85.3 98.9 93.2</td><td>99.0 99.3 93.0 98.9 98.8</td><td>96.7 97.5 98.2 98.9 99.1</td><td>96.4 97.6 96.2 97.8 100.0</td><td>96.3 97.8 99.3 99.5 100.0 100.0 100.0 100.0 97.6 100.0</td></tr><tr><td>NH∧S</td><td>CIFAR10 CIFAR100 LSUN_CR Imagenet- Noise Uniform</td><td>95.4 94.5 95.6 94.7 96.4 96.8</td><td>91.4 92.0 91.8 93.1 93.1</td><td>95.9 95.6 95.3 95.7 97.1 96.5</td><td>97.9 96.8 97.6 96.1 97.9 99.0 97.7 97.8 98.2 96.2 95.6 100.0</td><td>97.9</td><td>95.9 97.1 94.8 96.7 96.5 97.2 95.1 96.8 82.7 98.0 97.8</td><td>95.2 94.8 97.1 97.3 95.8 100.0</td><td>100.0 100.0 100.0 100.0 97.8 100.0</td><td>100.0 100.0 100.0 100.0 97.4 100.0</td></tr><tr><td>CITIIII</td><td>SVHN CIFAR100 LSUN_CR Imagenet- Noise Uniform</td><td>95.8 87.3 91.9 87.5 96.5 96.8</td><td>81.9 78.6 81.3 78.4 79.9 81.0</td><td>92.3 87.3 90.8 88.2 88.9 89.9</td><td>90.3 88.2 82.9 92.0 87.7 90.3 96.6</td><td>83.9 89.9 84.0 81.8 73.0</td><td>96.7 87.5 82.8 93.3 88.1 84.1 97.6 94.4 98.8 100.0</td><td>91.5 89.2 100.0</td><td>93.7 98.8 86.9 95.3 91.2 98.6 86.5 94.7 94.8 97.3</td><td>98.2 94.2 98.2 93.3 97.0</td></tr><tr><td>CEIRIII0</td><td>SVHN CIFAR10 LSUN_CR Imagenet- Noise Uniform</td><td>78.8 78.6 81.0 80.8 73.4 93.3</td><td>59.2 58.9 59.4 59.2 58.7 62.0</td><td>80.4 73.3 74.2 76.0 65.9 29.8</td><td>83.2 76.3 81.6 78.2 67.5 36.6</td><td>75.9 69.3 79.8 73.9 73.6 100.0</td><td>81.3 79.5 81.4 81.3 76.8 93.5</td><td>77.5 59.9 79.7 70.8 90.6 94.3</td><td>73.9 77.2 78.0 79.5 62.9 100.0</td><td>98.8 93.5 81.6 95.4 83.8 86.9 99.1</td><td>100.0 94.2 80.2 95.9 81.4 94.6 100.0</td></tr></table>
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+ Certified robustness against adversarial noise: We sample uniform noise images as they are obviously out-distribution for all tasks and certify using Corollary 3.1 the largest ball around the uniform noise sample on which CCU attains at most 1.1· uniform confidence, that is $1 . 1 \%$ on CIFAR100 and $1 1 \%$ on all other datasets. We describe how to compute the radius of this ball in Appendix D. In principle it could be possible that the certified balls contain training or test images. In Appendix E we show that this is not the case. We construct adversarial noise samples for all OOD methods by maximizing the confidence/minimizing the score via a PGD attack with 500 steps and 50 random restarts on this ball. Further details of the attack can be found in Appendix C.2. In Table 1 we show the results of running this attack on the different models. We used 200 noise images and we report clean test error on the in-distribution, the success rate (SR) (fraction of adversarial noise points for which the confidence resp. score inside the ball is higher resp. lower than the median of the in-distribution’s confidence/score) and the AUC for the separation of the generated adversarial noise images and the in-distribution based on confidence/score. By construction, see Corollary 3.1, our method provably makes no overconfident predictions but we nevertheless run the attack on CCU as well. We note that only CCU performs perfectly on this task for all datasets - all other OOD methods fail at least on one dataset, most of them on all. We also see that ACET achieves very robust performance which may be expected as it does some kind of adversarial training for OOD detection. Nevertheless, even though they are very rare, high-confidence adversarial noise images for ACET can be found on SVHN, CIFAR10 and CIFAR100 and ACET has no guarantees. We illustrate the generated adversarial noise images for all methods in Figure 2.
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+ OOD performance: For each dataset and method we report the AUC for the binary classification problem of discriminating in- and out-distribution based on confidence resp. score. The results are shown in Table 2. The list of datasets we use for OOD detection can be seen in Table 2. LSUN_CR refers to only the classroom class of LSUN and Imagenet- is a subset of 10000 resized Imagenet validation images, that have no overlap with CIFAR10/CIFAR100 classes. The noise dataset was obtained as in Hein et al. (2019) by first shuffling the pixels of the test images in the in-distribution and then smoothing them by a Gaussian filter of uniformly random width, followed by a rescaling so that the images have full range. GrCIFAR10 refers to the images in CIFAR10 being grayscaled and resized to $2 8 \mathbf { x } 2 8$ and Uniform describes images sampled uniformly at random from $[ 0 , \dot { 1 } ] ^ { d }$ . We see that OE and CCU have the best OOD performance. MCD is worse than the base model which confirms the results found in Leibig et al. (2017) that MCD is not useful for OOD. DE outperforms EDL but is not much better than the baseline for CIFAR10 and CIFAR100. The performance of Maha is worse than what has been reported in Lee et al. (2018c) which can have two reasons. We just use their version where one uses the scores only from the last layer and we do not calibrate hyperparameters for each test set separately but just once on the Tiny Image dataset. Especially on CIFAR10 we found that the results depend strongly on the step size. The results of ACET, GAN and ODIN are mixed but clearly outperform the baseline. Comparing Table 1 and Table 2 we see that most models perform well when evaluating on uniform noise but fail when finding the worst case in a small neighborhood around the noise point. Thus we think that such worst-case analysis should become standard in OOD evaluation.
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+ # 5 CONCLUSION
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+ In Hein et al. (2019) it has recently been shown that ReLU networks produce arbitrarily highly confident predictions far away from the training data, which could only be resolved by a modification of the network architecture. With CCU we present such a modification which explicitly integrates a generative model and provably show that the resulting neural network produces close to uniform predictions far away from the training data. Moreover, CCU is the only OOD method which can guarantee low confidence predictions over a whole volume rather than just pointwise and we show that all other OOD methods fail in this worst-case setting. CCU achieves this without loss in test accuracy or OOD performance. In the future it would be interesting to use more powerful generative models for which one can also guarantee their behavior far away from the training data. This is currently not the case for VAEs and GANs (Nalisnick et al., 2019; Hendrycks et al., 2019).
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+ # ACKNOWLEDGMENTS
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+ The author acknowledge support from the BMBF through the Tübingen AI Center (FKZ: 01IS18039A) and by the DFG TRR 248, project number 389792660 and the DFG Excellence Cluster “Machine Learning -New Perspectives for Science”, EXC 2064/1, project number 390727645. The authors thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting Alexander Meinke.
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+ Antonio Torralba, Rob Fergus, and William T Freeman. 80 million tiny images: A large data set for nonparametric object and scene recognition. IEEE transactions on pattern analysis and machine intelligence, 30(11):1958–1970, 2008.
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+
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+ W. Wang, A. Wang, A. Tamar, X. Chen, and P. Abbeel. Safer classification by synthesis. preprint, arXiv:1711.08534v2, 2018.
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+
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+ E. Wong and J. Zico Kolter. Provable defenses against adversarial examples via the convex outer adversarial polytope. In ICML, 2018.
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+
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+ H. Xiao, K. Rasul, and R. Vollgraf. Fashion-MNIST: a novel image dataset for benchmarking machine learning algorithms. preprint, arXiv:1708.07747, 2017.
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+
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+ # A APPENDIX - PROOF OF THEOREM 3.1
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+
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+ Theorem 3.1. Let $( x _ { i } , y _ { i } ) _ { i = 1 } ^ { n }$ be the training set of the training distribution. We define the model for the conditional probability over the classes $y \in \{ 1 , \ldots , M \}$ given $x$ as
259
+
260
+ $$
261
+ \hat { p } ( y | x ) = \frac { \hat { p } ( y | x , i ) \hat { p } ( x | i ) + \frac { \lambda } { M } \hat { p } ( x | o ) } { \hat { p } ( x | i ) + \lambda \hat { p } ( x | o ) } ,
262
+ $$
263
+
264
+ where λ = pˆ(o) $\begin{array} { r } { \lambda = \frac { \hat { p } ( o ) } { \hat { p } ( i ) } > 0 } \end{array}$ and $M > 1$ . Further, let the model for the marginal density of the in-distribution ${ \hat { p } } ( x | i )$ and out-distribution $p ( x | o )$ be given by the generalized GMMs
265
+
266
+ $$
267
+ \hat { p } ( x | i ) = \sum _ { k = 0 } ^ { K _ { i } } \alpha _ { k } \exp \left( - \frac { d ( x , \mu _ { k } ) ^ { 2 } } { 2 \sigma _ { k } ^ { 2 } } \right) , \qquad \hat { p } ( x | o ) = \sum _ { l = 0 } ^ { K _ { o } } \beta _ { l } \exp \left( - \frac { d ( x , \nu _ { l } ) ^ { 2 } } { 2 \theta _ { l } ^ { 2 } } \right)
268
+ $$
269
+
270
+ with $\alpha _ { k } , \beta _ { l } > 0$ and $\mu _ { k } , \nu _ { l } \in \mathbb { R } ^ { d } \quad \forall k = 1 , . . . K _ { i }$ , $l = 1 , \ldots , K _ { o }$ and $d : \mathbb { R } ^ { d } \times \mathbb { R } ^ { d } \mathbb { R } _ { + }$ a metric. Let $ { \boldsymbol { z } } _ { \mathbf { \lambda } } \in { \mathcal { \mathbf { R } } } ^ { d }$ and define $\begin{array} { r l r l r l } { k ^ { * } } & { { } = } & { \underset { k = 1 , \ldots , K _ { i } } { \arg \operatorname* { m i n } } \frac { d ( z , \mu _ { k } ) } { \sigma _ { k } } , } & { i ^ { * } } & { { } = } & { \underset { i = 1 , \ldots , n } { \arg \operatorname* { m i n } } d ( z , x _ { i } ) , } \end{array}$ $\operatorname * { a r g m i n } _ { l = 1 , \dots , K _ { o } } \beta _ { l } \exp { \left( - \frac { d ( z , \nu _ { l } ) ^ { 2 } } { 2 \theta _ { l } ^ { 2 } } \right) }$ and $\begin{array} { r } { \Delta = \frac { \theta _ { l ^ { * } } ^ { 2 } } { \sigma _ { k ^ { * } } ^ { 2 } } - 1 } \end{array}$ . For any $\epsilon > 0$ , $i f \operatorname* { m i n } _ { l } \theta _ { l } > \operatorname* { m a x } _ { k } \sigma _ { k }$ and
271
+
272
+ $$
273
+ \operatorname* { m i n } _ { i = 1 , \ldots , n } d ( z , x _ { i } ) \geq d ( x _ { i ^ { * } } , \mu _ { k ^ { * } } ) + d ( \mu _ { k ^ { * } } , \nu _ { l ^ { * } } ) \Big [ \frac { 2 } { \Delta } + \frac { 1 } { \sqrt { \Delta } } \Big ] + \theta _ { l ^ { * } } \sqrt { \frac { 2 } { \Delta } \log \Big ( \frac { M - 1 } { \epsilon \lambda } \frac { \sum _ { k } \alpha _ { k } } { \beta _ { l ^ { * } } } \Big ) } ,
274
+ $$
275
+
276
+ then it holds for all $m \in \{ 1 , \ldots , M \}$ that
277
+
278
+ $$
279
+ \hat { p } ( m | z ) \leq \frac { 1 } { M } \big ( 1 + \epsilon \big ) .
280
+ $$
281
+
282
+ In particular, $i f \operatorname* { m i n } _ { i } d ( z , x _ { i } ) \to \infty$ , then $\begin{array} { r } { \hat { p } ( m | z ) \to \frac { 1 } { M } } \end{array}$ .
283
+
284
+ Proof. The proof essentially hinges on upper bounding $\frac { { \hat { p } } ( z | i ) } { { \hat { p } } ( z | o ) }$ using the specific properties of the Gaussian mixture model. We note that
285
+
286
+ $$
287
+ \hat { p } ( y | x ) = \frac { \hat { p } ( y | x , i ) \hat { p } ( x | i ) + \frac { \lambda } { M } \hat { p } ( x | o ) } { \hat { p } ( x | i ) + \lambda \hat { p } ( x | o ) } = \frac { 1 } { M } \frac { 1 + \frac { M } { \lambda } \frac { \hat { p } ( x | i ) } { \hat { p } ( x | o ) } } { 1 + \frac { 1 } { \lambda } \frac { \hat { p } ( x | i ) } { \hat { p } ( x | o ) } } \leq \frac { 1 } { M } \left( 1 + \frac { M - 1 } { \lambda } \frac { \hat { p } ( x | i ) } { \hat { p } ( x | o ) } \right)
288
+ $$
289
+
290
+ The last step holds because the function $\begin{array} { r } { g ( \xi ) = \frac { 1 + M \xi } { 1 + \xi } } \end{array}$ is monotonically increasing
291
+
292
+ $$
293
+ \frac { \partial g } { \partial \xi } = \frac { M - 1 } { ( 1 + \xi ) ^ { 2 } } ~ \mathrm { a n d } ~ \frac { \partial ^ { 2 } g } { \partial \xi ^ { 2 } } = - 2 \frac { M - 1 } { ( 1 + \xi ) ^ { 3 } } .
294
+ $$
295
+
296
+ As the second deriviative is negative for $\xi \ge 0$ , $g$ is concave for $\xi \ge 0$ and thus
297
+
298
+ $$
299
+ \frac { 1 + M \xi } { 1 + \xi } = g ( \xi ) \leq g ( 0 ) + \frac { \partial g } { \partial \xi } \Big | _ { \xi = 0 } ( \xi - 0 ) = 1 + ( M - 1 ) \xi .
300
+ $$
301
+
302
+ In order to achieve the required result we need to show that $\begin{array} { r } { \frac { M - 1 } { \lambda } \frac { \hat { p } ( x | i ) } { \hat { p } ( x | o ) } \le \epsilon } \end{array}$ for $x$ sufficiently far away from the training data.
303
+
304
+ We note that
305
+
306
+ $$
307
+ \begin{array} { r l r } { \frac { \hat { p } ( x | i ) } { \hat { p } ( x | o ) } = \frac { \sum _ { k } \alpha _ { k } \exp { ( - \frac { d ( x , \mu _ { k } ) ^ { 2 } } { 2 \sigma _ { k } ^ { 2 } } ) } } { \sum _ { l } \beta _ { l } \exp { ( - \frac { d ( x , \nu _ { l } ) ^ { 2 } } { 2 \theta _ { l } ^ { 2 } } ) } } \leq \frac { \operatorname* { m a x } _ { k } \exp { ( - \frac { d ( x , \mu _ { k } ) ^ { 2 } } { 2 \sigma _ { k } ^ { 2 } } ) } \sum _ { k } \alpha _ { k } } { \operatorname* { m a x } _ { l } \beta _ { l } \exp { ( - \frac { d ( x , \nu _ { l } ) ^ { 2 } } { 2 \theta _ { l } ^ { 2 } } ) } } } \\ & { } & { = \frac { \sum _ { k } \alpha _ { k } } { \beta _ { l ^ { * } } } \exp { ( - \frac { d ( x , \mu _ { k ^ { * } } ) ^ { 2 } } { 2 \sigma _ { k ^ { * } } ^ { 2 } } + \frac { d ( x , \nu _ { l ^ { * } } ) ^ { 2 } } { 2 \theta _ { l ^ { * } } ^ { 2 } } ) } } \end{array}
308
+ $$
309
+
310
+ where k∗ = arg min d(x,µk)22σ2 and $l ^ { * } = \arg \operatorname* { m a x } _ { l } \beta _ { l } \exp ( - \frac { d ( x , \nu _ { l } ) ^ { 2 } } { 2 \theta _ { l } ^ { 2 } } )$ Using the triangle inequality, $d ( x , \nu _ { l ^ { * } } ) \leq d ( x , \mu _ { k ^ { * } } ) + d ( \mu _ { k ^ { * } } , \nu _ { l ^ { * } } )$ , we get the desired condition as
311
+
312
+ $$
313
+ \frac { \sum _ { k } \alpha _ { k } } { \beta _ { l ^ { * } } } \exp \left( - d ( x , \mu _ { k ^ { * } } ) ^ { 2 } \left( \frac { 1 } { 2 \sigma _ { k ^ { * } } ^ { 2 } } - \frac { 1 } { 2 \theta _ { l ^ { * } } ^ { 2 } } \right) + \frac { d ( \mu _ { k ^ { * } } , \nu _ { l ^ { * } } ) d ( x , \mu _ { k ^ { * } } ) } { \theta _ { l ^ { * } } ^ { 2 } } + \frac { d ( \mu _ { k ^ { * } } , \nu _ { l ^ { * } } ) ^ { 2 } } { 2 \theta _ { l ^ { * } } ^ { 2 } } \right) \leq \frac { \epsilon \lambda } { M - 1 }
314
+ $$
315
+
316
+ Thus we get with a = 12σ2k∗ $\begin{array} { r } { a = \big ( \frac { 1 } { 2 \sigma _ { k ^ { * } } ^ { 2 } } - \frac { 1 } { 2 \theta _ { l ^ { * } } ^ { 2 } } \big ) } \end{array}$ , $\begin{array} { r } { b = \frac { d ( \mu _ { k ^ { * } } , \nu _ { l ^ { * } } ) } { \theta _ { l ^ { * } } ^ { 2 } } } \end{array}$ and $\begin{array} { r } { c = \frac { d ( \mu _ { k ^ { * } } , \nu _ { l ^ { * } } ) ^ { 2 } } { 2 \theta _ { l ^ { * } } ^ { 2 } } } \end{array}$ , $\begin{array} { r } { d = \log \left( \frac { \epsilon \lambda } { M - 1 } \frac { \beta _ { l ^ { * } } } { \sum _ { k } \alpha _ { k } } \right) } \end{array}$ , the quadratic inequality
317
+
318
+ $$
319
+ - d ( x , \mu _ { k ^ { * } } ) ^ { 2 } a + d ( x , \mu _ { k ^ { * } } ) b + c \leq d ,
320
+ $$
321
+
322
+ where $d < 0$ for sufficiently small $\epsilon$ . We get the solution
323
+
324
+ $$
325
+ d ( x , \mu _ { k ^ { * } } ) \geq \frac { b } { 2 a } + \sqrt { \operatorname* { m a x } \left\{ 0 , \frac { c - d } { a } + \frac { b ^ { 2 } } { 4 a ^ { 2 } } \right\} } .
326
+ $$
327
+
328
+ It holds, using ${ \sqrt { a + b } } \leq { \sqrt { a } } + { \sqrt { b } }$ for $a , b > 0$ ,
329
+
330
+ $$
331
+ \frac { b } { 2 a } + \sqrt { \operatorname* { m a x } \left\{ 0 , \frac { c - d } { a } + \frac { b ^ { 2 } } { 4 a ^ { 2 } } \right\} } \leq \frac { b } { a } + \sqrt { \frac { c } { a } } + \sqrt { \frac { - d } { a } } .
332
+ $$
333
+
334
+ One can simplify
335
+
336
+ $$
337
+ \begin{array} { r } { \frac { b } { a } = 2 \frac { \sigma _ { k ^ { * } } ^ { 2 } \theta _ { l ^ { * } } ^ { 2 } } { \theta _ { l ^ { * } } ^ { 2 } - \sigma _ { k ^ { * } } ^ { 2 } } \frac { d ( \mu _ { k ^ { * } } , \nu _ { l ^ { * } } ) } { \theta _ { l ^ { * } } ^ { 2 } } = 2 \frac { \sigma _ { k ^ { * } } ^ { 2 } d ( \mu _ { k ^ { * } } , \nu _ { l ^ { * } } ) } { \theta _ { l ^ { * } } ^ { 2 } - \sigma _ { k ^ { * } } ^ { 2 } } = 2 \frac { d ( \mu _ { k ^ { * } } , \nu _ { l ^ { * } } ) } { \frac { \theta _ { l ^ { * } } ^ { 2 } } { \sigma _ { k ^ { * } } ^ { 2 } } - 1 } } \\ { \frac { c } { a } = 2 \frac { \sigma _ { k ^ { * } } ^ { 2 } \theta _ { l ^ { * } } ^ { 2 } } { \theta _ { l ^ { * } } ^ { 2 } - \sigma _ { k ^ { * } } ^ { 2 } } \frac { d ( \mu _ { k ^ { * } } , \nu _ { l ^ { * } } ) ^ { 2 } } { 2 \theta _ { l ^ { * } } ^ { 2 } } = \frac { \sigma _ { k ^ { * } } ^ { 2 } d ( \mu _ { k ^ { * } } , \nu _ { l ^ { * } } ) ^ { 2 } } { \theta _ { l ^ { * } } ^ { 2 } - \sigma _ { k ^ { * } } ^ { 2 } } = \frac { d ( \mu _ { k ^ { * } } , \nu _ { l ^ { * } } ) ^ { 2 } } { \frac { \theta _ { l ^ { * } } ^ { 2 } } { \sigma _ { k ^ { * } } ^ { 2 } } - 1 } } \end{array}
338
+ $$
339
+
340
+ Noting that $d ( x , \mu _ { k ^ { * } } ) \geq | d ( x , x _ { i ^ { * } } ) - d ( x _ { i ^ { * } } , \mu _ { k ^ { * } } ) |$ we get that
341
+
342
+ $$
343
+ d ( x , x _ { i ^ { * } } ) \geq d ( x _ { i ^ { * } } , \mu _ { k ^ { * } } ) + { \frac { b } { a } } + { \sqrt { \frac { c } { a } } } + { \sqrt { \frac { - d } { a } } } ,
344
+ $$
345
+
346
+ implies M−1 pˆ(x|i) $\begin{array} { r } { \frac { M - 1 } { \lambda } \frac { \hat { p } ( x | i ) } { \hat { p } ( x | o ) } \le \epsilon } \end{array}$ . The last statement follows directly by noting that by assumption $a > 0$ (independently of the choice of $l ^ { * }$ and $k ^ { * }$ ) and $b , c , d ( x _ { i ^ { * } } , \mu _ { k ^ { * } } )$ are bounded as $K _ { i } , K _ { o } , n$ are finite. With $\begin{array} { r } { \Delta = \frac { \theta _ { l ^ { * } } ^ { 2 } } { \sigma _ { k ^ { * } } ^ { 2 } } - 1 } \end{array}$ we can rewrite the required condition as
347
+
348
+ $$
349
+ d ( x , x _ { i ^ { * } } ) \geq d ( x _ { i ^ { * } } , \mu _ { k ^ { * } } ) + d ( \mu _ { k ^ { * } } , \nu _ { l ^ { * } } ) \Big [ \frac { 2 } { \Delta } + \frac { 1 } { \sqrt { \Delta } } \Big ] + \theta _ { l ^ { * } } \sqrt { \frac { 2 } { \Delta } \log \Big ( \frac { M - 1 } { \epsilon \lambda } \frac { \sum _ { k } \alpha _ { k } } { \beta _ { l ^ { * } } } \Big ) } .
350
+ $$
351
+
352
+ # B APPENDIX - PROOF OF COROLLARY 3.1
353
+
354
+ Corollary 3.1. Let $x _ { 0 } \in \mathbb { R } ^ { d }$ and $R > 0 ;$ , then with $\begin{array} { r } { \lambda = \frac { \hat { p } ( o ) } { \hat { p } ( i ) } } \end{array}$ it holds
355
+
356
+ where
357
+
358
+ $$
359
+ \begin{array} { r l r } & { } & { \underset { d ( x , x _ { 0 } ) \leq R } { \operatorname* { m a x } } \ \hat { p } ( y | x ) \leq \frac { 1 } { M } \frac { 1 + M \frac { b } { \lambda } } { 1 + \frac { b } { \lambda } } , } \\ & { } & { \quad \frac { \sum _ { k = 1 } ^ { K _ { i } } \alpha _ { k } \exp { \left( - \frac { \operatorname* { m a x } \left\{ d ( x _ { 0 } , \mu _ { k } ) - R , 0 \right\} ^ { 2 } } { 2 \sigma _ { k } ^ { 2 } } \right) } } { \sum _ { l = 1 } ^ { K _ { o } } \beta _ { l } \exp { \left( - \frac { ( d ( x _ { 0 } , \nu _ { l } ) + R ) ^ { 2 } } { 2 \theta _ { l } ^ { 2 } } \right) } } . } \end{array}
360
+ $$
361
+
362
+ Proof. From the previous section we already know that pˆ(y|x) ≤ 1M 1 as long as $\begin{array} { r } { \frac { p ( x | i ) } { p ( x | o ) } \leq b } \end{array}$ .
363
+ Now we can separately bound the numerator and denominator within a ball of radius $R$ around $x _ { 0 }$ .
364
+
365
+ For the numerator we have
366
+
367
+ $$
368
+ \begin{array} { r l } { \displaystyle \operatorname* { m a x } _ { d ( x , x _ { 0 } ) \leq R } \hat { p } ( x | i ) \leq \sum _ { k = 1 } ^ { K } \alpha _ { k } \operatorname* { m a x } _ { d ( x , x _ { 0 } ) \leq R } e ^ { - \frac { d ( x , \mu _ { k } ) ^ { 2 } } { 2 \sigma _ { k } ^ { 2 } } } } & { } \\ { \leq \sum _ { k = 1 } ^ { K } \alpha _ { k } \exp \left( - \frac { \operatorname* { m i n } _ { \operatorname* { m i n } } d ( x , \mu _ { k } ) ^ { 2 } } { 2 \sigma _ { k } ^ { 2 } } \right) } & { } \\ { \leq \sum _ { k = 1 } ^ { K } \alpha _ { k } \exp \left( - \frac { \left( \operatorname* { m a x } \left\{ d ( \mu _ { k } , x _ { 0 } ) - R , 0 \right\} \right) ^ { 2 } } { 2 \sigma _ { k } ^ { 2 } } \right) , } \end{array}
369
+ $$
370
+
371
+ where we have lower bounded $\operatorname* { m i n } _ { d ( x , x _ { 0 } ) \leq R } d ( x , \mu _ { k } )$ via the reverse triangle inequality
372
+
373
+ $$
374
+ \begin{array} { l l } { \displaystyle \operatorname* { m i n } _ { d ( x , x _ { 0 } ) \leq R } d ( x , \mu _ { k } ) \geq \operatorname* { m i n } _ { d ( x , x _ { 0 } ) \leq R } \lvert d ( x _ { 0 } , \mu _ { k } ) - d ( x , x _ { 0 } ) \rvert , } \\ { \geq \operatorname* { m a x } \left\{ \operatorname* { m i n } _ { d ( x , x _ { 0 } ) \leq R } ( d ( x _ { 0 } , \mu _ { k } ) - d ( x _ { 0 } , \mu _ { k } ) ) , 0 \right\} , } \\ { \geq \operatorname* { m a x } \left\{ d ( x _ { 0 } , \mu _ { k } ) - r , 0 \right\} . } \end{array}
375
+ $$
376
+
377
+ The denominator can similarly be bounded via
378
+
379
+ $$
380
+ \begin{array} { r l r } { { \operatorname* { m i n } _ { d ( x , x _ { 0 } ) \leq R } \hat { p } ( x | o ) \geq \sum _ { l = 1 } ^ { K _ { o } } \beta _ { l } \operatorname* { m i n } _ { d ( x , x _ { 0 } ) \leq R } e ^ { - \frac { d ( x , \nu _ { l } ) ^ { 2 } } { 2 \theta _ { k } ^ { 2 } } } } } \\ & { } & { \geq \sum _ { l = 1 } ^ { K _ { o } } \beta _ { l } \exp ( - \frac { d ( x , x _ { 0 } ) } { 2 \theta _ { l } ^ { 2 } } \Delta \theta _ { l } ^ { 2 } ) } \\ & { } & { \geq \sum _ { l = 1 } ^ { K _ { o } } \beta _ { l } \exp ( - \frac { ( d ( x _ { 0 } , \nu _ { l } ) + R ) ^ { 2 } } { 2 \theta _ { l } ^ { 2 } } ) . } \end{array}
381
+ $$
382
+
383
+ With both of these bounds in place the conclusion immediately follows.
384
+
385
+ # C APPENDIX - EXPERIMENTAL DETAILS
386
+
387
+ Unless specified otherwise we use ADAM on MNIST with a learning rate of $1 e - 3$ and SGD with learning rate 0.1 for the other datasets. The learning rate for the GMM is always set to $1 e - 5$ . We decrease all learning rates by a factor of 10 after 50, 75 and 90 epochs. Our batch size is 128, the total number of epochs 100 and weight decay is set to $5 e - 4$ .
388
+
389
+ When training ACET, OE and CCU with 80 million tiny images we pick equal batches of in- and out-distribution data (corresponding to $p ( i ) = p ( o ) ,$ ) and concatenate them into a batches of size 256. Note that during the 100 epochs only a fraction of the 80 million tiny images are seen and so there is no risk of over-fitting.
390
+
391
+ # C.1 DATA AUGMENTATION
392
+
393
+ Our data augmentation scheme uses random crops with a padding of 2 pixels on MNIST and FMNIST. On SVHN, CIFAR10 and CIFAR100 the padding width is 4 pixels. For SVHN we fill the padding with the value at the boundary and for CIFAR we apply reflection at the boundary pixels. On top of this we include random horizontal flips on CIFAR. For MNIST and FMNIST we generate 60000 such samples and for SVHN and CIFAR 50000 samples by drawing from the clean dataset without replacement. This augmented data is used to calculate the covariance matrix from equation 12. During the actual training we use the same data augmentation scheme in a standard fashion.
394
+
395
+ # C.2 ATTACK DETAILS
396
+
397
+ We begin with a step size of 3 and for each of the 50 restarts we randomly initialize at some point in the ellipsoid. Whenever a gradient step successfully decreases the losses we increase the step size by a factor of 1.1. Whenever the loss increases instead we use backtracking and decrease the step size by a factor of 2. We apply normal PGD using the $l _ { 2 }$ -norm in the transformed space to ensure that we stay on the ellipsoid and after each gradient step we rotate back into the original space to project onto the box $[ 0 , 1 ] ^ { d }$ . The result is not guaranteed be on the ellipsoid so after the 500 steps we use the alternating projection algorithm (Bauschke & Borwein, 1996) for 10 steps which is guaranteed to converge to a point in the intersection of the ellipsoid and the box because both of these sets are convex.
398
+
399
+ ![](images/c0433f8203a9b24e6f0864c0cdccf0540eb88f99063b38c9fa7a4f0b8af0fd29.jpg)
400
+ Figure 3: Histograms of bounds: Certified radius in transformed space for different datasets.
401
+
402
+ # D APPENDIX - FINDING A THE CERTIFIABLE RADIUS
403
+
404
+ Since Corollary 3.1 does not explicitly give a radius, one has to numerically invert the bound. The bound
405
+
406
+ $$
407
+ b ( R ) = \frac { \sum _ { k = 1 } ^ { K _ { i } } \alpha _ { k } \exp { \left( - \frac { \operatorname* { m a x } \{ d ( x _ { 0 } , \mu _ { k } ) - R , 0 \} ^ { 2 } } { 2 \sigma _ { k } ^ { 2 } } \right) } } { \sum _ { l = 1 } ^ { K _ { o } } \beta _ { l } \exp { \left( - \frac { ( d ( x _ { 0 } , \nu _ { l } ) + R ) ^ { 2 } } { 2 \theta _ { l } ^ { 2 } } \right) } }
408
+ $$
409
+
410
+ is monotonically increasing in $R$ . Thus, for a given sample $x _ { 0 }$ one can fix a desired bound $\operatorname* { m a x } _ { d ( x , x _ { 0 } ) \leq R } \hat { p } ( x | i ) \leq \frac { 1 } { M } \nu$ , where $\nu \in ( 1 , M )$ and then find the unique solution
411
+
412
+ $$
413
+ b ( R ) = \frac { \nu - 1 } { M - \nu } \lambda
414
+ $$
415
+
416
+ for $R$ via bisection. This radius $\hat { R }$ will then represent the maximal radius, that one can certify using Corollary 3.1. The presumption is, of course, that for $R = 0$ one has a sufficiently low bound in the first place, i.e. that a solution exists. In our experiments on uniform noise we did not encounter a single counterexample to this assumption. We show the radii for the different datasets in Figure 3.
417
+
418
+ # E APPENDIX - ANALYSIS OF THE CERTIFIED BALLS AROUND UNIFORM NOISE IMAGES
419
+
420
+ As one can observe in Figure 2 the images which maximize the confidence in the certified ball around the uniform noise image are sometimes quite far away from the original noise image. As CCU certifies low confidence (the maximal confidence is less than $1 . 1 \times { \frac { 1 } { M } }$ - so the predicted probability distribution over the classes is very close to the uniform distribution) over the whole ball, it is a natural question what these balls look like and what kind of images they contain. In particular, it is in general not desired that the certified balls contain images from the training and test set. For each dataset we certified balls around 200 uniform noise images and for each of the certified balls we check if it contains training or test images of the corresponding dataset. We found that even though the certified balls are large, not a single training or test image was contained in any of them. This justifies the use of our proposed threat model.
421
+
422
+ A different problem could be that our threshold of $\textstyle { \frac { 1 . 1 } { M } }$ for the certification is too high and that many predictions on the test set have confidence less than this threshold. For this purpose we report in Table 3 the smallest predicted confidence of CCU on the test set $T$ , that is
423
+
424
+ $$
425
+ \operatorname* { m i n } _ { x \in T } \operatorname* { m a x } _ { y \in \{ 1 , . . . , M \} } \hat { p } ( y | x ) ,
426
+ $$
427
+
428
+ Table 3: Lowest confidence that CCU attains on the test set (in percent) as well as total number of test points on which confidence is lower than our imposed bound of $\textstyle { \frac { 1 . 1 } { M } }$ .
429
+
430
+ <table><tr><td></td><td>min p(ylx) #&lt; 1.1 %&lt; M</td></tr><tr><td>MNIST</td><td>33.08 0</td></tr><tr><td>28.77</td><td>0 0 0</td></tr><tr><td>FMNIST SVHN</td><td></td></tr><tr><td>10.02 20 0.08</td><td></td></tr><tr><td>CIFAR10 10.01 CIFAR100 1.03 130</td><td>529 5.29 1.30</td></tr></table>
431
+
432
+ for each dataset and the total number of test samples where the confidence is below $\textstyle { \frac { 1 . 1 } { M } }$ . While for MNIST and FMNIST, this never happens, and for SVHN this is negligible (less than $0 . \dot { 1 } \%$ of the test set), for CIFAR10 and CIFAR100 this happens in $5 . 3 \%$ resp. $1 . 3 \%$ of the cases.
433
+
434
+ In theory, this could impair our AUC value for the detection of adversarial noise. However, in practice our bound for the confidence is quite conservative as the bound is only tight in very specific configurations of the centroids of the Gaussian mixture model which are unlikely to happen for any practical dataset, meaning that the actual maximal confidence in the certified region is typically significantly lower. In fact the AUC values of CCU are always $1 0 0 \%$ which means that for all 200 certified balls the maximal value of the confidence of CCU in any of these balls (found by our PGD attack algorithm) is lower than the minimal confidence of all predictions on the test set as reported in Table 3. On the other hand assuming here also a worst case scenario in the sense we assume that the upper bound of the maximal confidence is attained in all 200 certified balls, then the (certified) AUC value would be: $9 9 . 9 2 \%$ for SVHN, $9 4 . 7 1 \%$ for CIFAR10, and $9 8 . 7 0 \%$ for CIFAR100. Note that this theoretical lower bound on our performance is still better than all other models’ empirical performance on this task, as reported in Table 1 on both CIFAR10 and CIFAR100, and only marginally below the perfect AUC of GAN on SVHN.
435
+
436
+ # F APPENDIX - PERFORMANCE OF ACET WHEN TRAINED ON ADVERSARIAL UNIFORM NOISE
437
+
438
+ Similar to our CCU, ACET Hein et al. (2019) requires that one chooses a model for the out-distribution in order to generate their “adversarial noise” during training. We trained the ACET model with the same out-distribution model as for all other models namely using the tiny image dataset as suggested in Hendrycks et al. (2019) with a PGD attack that starts at the original point and takes 40 FGSM steps in order to maximize the maximal confidence over the classes. We use backtracking and halve the step size whenever the loss does not increase. However, the authors of Hein et al. (2019) used smoothed uniform noise and a $l _ { \infty }$ -threat model during training. Since our worst case analysis for OOD is based on attacking uniform noise images, this suggests that training ACET with uniform noise should improve the performance of ACET for the worst case analysis. We report below the results of ACET2 (the original model in the paper using tiny images for training is called ACET) based on attacking using uniform noise images during training with a $l _ { \infty }$ -threat model with $\epsilon = 0 . 3$ as suggested in Hein et al. (2019). We report the normal OOD performance in Table 4 and the worst case analysis of adversarial noise in Table 5. While it is not surprising that ACET outperforms ACET2 on the standard OOD detection task in Table 4 as it has seen more realistic “noise” images during training, the worse performance of ACET2 for the worst case analysis in Table 5 is at first sight counter-intuitive. However, note that the threat model of the attacks in our worst case analysis is the Mahalanobis-type $l _ { 2 }$ -type metric, see 12, while ACET2 uses an $l _ { \infty }$ -attack model with $\epsilon = 0 . 3$ during training. As the size of the balls for the Mahalanobis-type $l _ { 2 }$ -type metric is quite large, there is not much overlap between the two sets. This explains why ACET2 fails here. In summary, we have shown that by using tiny images as out-distribution during training, ACET improves in terms of OOD detection performance over ACET2, which is similar to the version suggested in Hein et al. (2019).
439
+
440
+ # G APPENDIX - PRECISION AND RECALL
441
+
442
+ In addition to the AUC presented in Table 2 we follow Hendrycks & Gimpel (2017c) and report the area under the precision/recall curve (AUPR). Precision at a specific threshold is defined as the
443
+
444
+ <table><tr><td>MNIST</td><td>ACET</td><td>ACET2</td></tr><tr><td>FMNIST</td><td>100.0</td><td>99.8</td></tr><tr><td>EMNIST</td><td>95.0</td><td>93.5</td></tr><tr><td>GrCIFAR10</td><td>100.0</td><td>100.0</td></tr><tr><td>Noise</td><td>100.0</td><td>100.0</td></tr><tr><td>Uniform</td><td>100.0</td><td>100.0</td></tr></table>
445
+
446
+ <table><tr><td>FMNIST</td><td>ACET</td><td>ACET2</td></tr><tr><td>MNIST</td><td>96.4</td><td>96.5</td></tr><tr><td>EMNIST</td><td>97.6</td><td>97.3</td></tr><tr><td>GrCIFAR10</td><td>96.2</td><td>91.6</td></tr><tr><td>Noise</td><td>97.8</td><td>97.1</td></tr><tr><td>Uniform</td><td>100.0</td><td>100.0</td></tr></table>
447
+
448
+ <table><tr><td>SVHN</td><td>ACET</td><td>ACET2</td><td>CIFAR10</td><td>ACET</td><td>ACET2</td></tr><tr><td>CIFAR10</td><td>95.2</td><td>94.2</td><td>SVHN</td><td>93.7</td><td>82.8</td></tr><tr><td>CIFAR100</td><td>94.8</td><td>93.7</td><td>CIFAR100</td><td>86.9</td><td>85.3</td></tr><tr><td>LSUN_CR</td><td>97.1</td><td>96.1</td><td>LSUN_CR</td><td>91.2</td><td>88.5</td></tr><tr><td>Imagenet-</td><td>97.3</td><td>95.6</td><td>Imagenet-</td><td>86.5</td><td>84.8</td></tr><tr><td>Noise</td><td>95.2</td><td>95.2</td><td>Noise</td><td>94.8</td><td>91.2</td></tr><tr><td>Uniform</td><td>100.0</td><td>100.0</td><td>Uniform</td><td>100.0</td><td>100.0</td></tr></table>
449
+
450
+ Table 4: OOD detection performance (AUC in percent) for ACET (trained around tiny images) and ACET2 (trained around uniform noise).
451
+
452
+ <table><tr><td>CIFAR100</td><td>ACET</td><td>ACET2</td></tr><tr><td>SVHN</td><td>73.9</td><td>84.6</td></tr><tr><td>CIFAR10</td><td>77.2</td><td>77.0</td></tr><tr><td>LSUN_CR</td><td>78.0</td><td>80.0</td></tr><tr><td>Imagenet-</td><td>79.5</td><td>79.4</td></tr><tr><td>Noise</td><td>62.9</td><td>66.3</td></tr><tr><td>Uniform</td><td>100.0</td><td>100.0</td></tr></table>
453
+
454
+ <table><tr><td colspan="2"></td><td>ACET</td><td>ACET2</td></tr><tr><td rowspan="3">MNIST</td><td>TE</td><td>0.6</td><td>0.6</td></tr><tr><td>SR</td><td>0.0</td><td>0.0</td></tr><tr><td>AUC</td><td>100.0</td><td>100.0</td></tr><tr><td rowspan="3">FMNIST</td><td>TE</td><td>4.8</td><td>4.6</td></tr><tr><td>SR</td><td>0.0</td><td>0.0</td></tr><tr><td>AUC</td><td>100.0</td><td>100.0</td></tr><tr><td rowspan="3">SVHN</td><td>TE SR</td><td>3.2</td><td>3.0</td></tr><tr><td></td><td>3.0</td><td>95.5</td></tr><tr><td>AUC</td><td>96.5</td><td>5.4</td></tr><tr><td rowspan="3">CIFAR10</td><td>TE</td><td>6.1</td><td>7.1</td></tr><tr><td>SR</td><td>0.0</td><td>64.5</td></tr><tr><td>AUC</td><td>99.9</td><td>35.9</td></tr><tr><td rowspan="3">CIFAR100</td><td>TE</td><td>25.2</td><td>26.2</td></tr><tr><td>SR</td><td>3.5</td><td>96.5</td></tr><tr><td>AUC</td><td>95.8</td><td>14.6</td></tr></table>
455
+
456
+ Table 6: AUPR (in- versus out-distribution detection based on confidence/score) in percent for different OOD methods and datasets (higher is better). OE and CCU have the best OOD performance.
457
+
458
+ <table><tr><td colspan="2"></td><td>Base</td><td>MCD</td><td>EDL</td><td>DE</td><td>GAN</td><td>ODIN</td><td>Maha</td><td>ACET</td><td>OE</td><td>CCU</td></tr><tr><td rowspan="4">LSINΛ</td><td>FMNIST</td><td>97.5</td><td>89.4</td><td>99.4 77.3</td><td>99.4 84.5</td><td>99.4 85.5</td><td>98.8 78.4</td><td>97.0 74.4</td><td>100.0 90.9</td><td>99.9 91.4</td><td>99.9 84.3</td></tr><tr><td rowspan="4">EMNIST GrCIFAR10 Noise</td><td>77.9 99.7</td><td>60.0</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>91.1</td><td>99.8</td><td>100.0</td><td>99.5</td><td>99.9</td><td>98.9</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>100.0 97.2</td><td>75.5 82.8</td><td>99.8 99.9</td><td>100.0 98.8</td><td>99.2</td><td>100.0</td><td>96.5</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td>Uniform MNIST</td><td>97.6</td><td>79.3</td><td>95.9</td><td>97.5</td><td>99.9 99.9</td><td>98.9 99.2</td><td>100.0</td><td>100.0</td><td>100.0</td><td>100.0</td></tr><tr><td rowspan="5">JSINIH</td><td>EMNIST GrCIFAR10</td><td>96.8</td><td>74.2</td><td>94.6</td><td>96.1</td><td>100.0</td><td>98.9</td><td>97.2 96.5</td><td>97.4 97.0</td><td>97.0 98.6</td><td>98.3 99.1</td></tr><tr><td rowspan="5">Noise Uniform</td><td>92.2</td><td>92.7</td><td>86.9</td><td>90.8</td><td>82.3</td><td>92.9</td><td>98.6</td><td>96.8</td><td>100.0</td><td>100.0</td></tr><tr><td>93.8</td><td>78.6</td><td>91.6</td><td>92.5</td><td>95.4</td><td>95.4</td><td>97.0</td><td>95.3</td><td>100.0</td><td>100.0</td></tr><tr><td>97.8</td><td>93.0</td><td>97.1</td><td>98.8</td><td>95.4</td><td>99.1</td><td></td><td></td><td></td><td></td></tr><tr><td>97.2</td><td></td><td></td><td></td><td></td><td></td><td>99.4</td><td>100.0</td><td>98.2</td><td>100.0</td></tr><tr><td rowspan="5">CIFAR10 CIFAR100 NH∧S Imagenet- Noise</td><td>96.7</td><td>96.2 95.8</td><td>98.5 98.3</td><td>99.2 99.0</td><td>98.6 98.2</td><td>97.3 96.6</td><td>99.0</td><td>97.3</td><td>100.0</td><td>100.0</td></tr><tr><td>LSUN_CR</td><td>99.9 99.9</td><td>99.9</td><td>100.0</td><td>100.0</td><td>99.9</td><td>98.8 100.0</td><td>97.0 100.0</td><td>100.0 100.0</td><td>100.0 100.0</td></tr><tr><td>96.8</td><td>96.2</td><td>98.3</td><td>99.1</td><td>98.9</td><td>96.9</td><td>98.9</td><td>98.3</td><td>100.0</td><td></td></tr><tr><td>89.1</td><td>83.6</td><td>95.6</td><td>96.8</td><td>94.5</td><td>50.3</td><td>97.0</td><td>87.3</td><td>95.6</td><td>100.0</td></tr><tr><td>98.5</td><td>97.0</td><td>98.7</td><td>98.5</td><td>100.0</td><td>98.9</td><td>99.3</td><td>100.0</td><td>100.0</td><td>93.3 100.0</td></tr><tr><td rowspan="5">CIIIIII Noise</td><td rowspan="5">SVHN CIFAR100 LSUN_CR Imagenet-</td><td>92.3</td><td>71.1</td><td>90.7</td><td>87.4</td><td>80.5</td><td>92.7</td><td>85.9</td><td>91.0</td><td>98.5</td><td>97.5</td></tr><tr><td>86.3</td><td>80.0</td><td>89.3</td><td>89.8</td><td>84.0</td><td>85.5</td><td>83.4</td><td>87.4</td><td>95.6</td><td>94.6</td></tr><tr><td>99.7</td><td>99.2</td><td>99.7</td><td>99.7</td><td>99.7</td><td>99.7</td><td>99.6</td><td>99.7</td><td>100.0</td><td>99.9</td></tr><tr><td>84.6</td><td>79.2</td><td>89.8</td><td>88.6</td><td>84.9</td><td>84.2</td><td>84.4</td><td>85.4</td><td>94.8</td><td>93.2</td></tr><tr><td>88.1 97.8</td><td>55.7</td><td>70.4</td><td>82.7</td><td>68.6</td><td>87.7</td><td>84.9</td><td>84.0</td><td>93.8</td><td>94.7</td></tr><tr><td rowspan="6">CEIRIIIO Noise Uniform</td><td rowspan="5">Uniform SVHN CIFAR10 LSUN_CR Imagenet-</td><td>67.4</td><td>83.4</td><td>94.5</td><td>98.0</td><td>82.4</td><td>99.1</td><td>100.0</td><td>100.0</td><td>99.2</td><td>100.0</td></tr><tr><td>80.9</td><td>52.9</td><td>72.0</td><td>75.6</td><td>63.4</td><td>71.0</td><td>69.1</td><td>59.4</td><td>89.6</td><td>90.8</td></tr><tr><td>99.3</td><td>66.1</td><td>75.8</td><td>79.1</td><td>72.5 99.2</td><td>81.3</td><td>61.2</td><td>79.7</td><td>84.8 99.9</td><td>83.7</td></tr><tr><td>81.9</td><td>98.3 67.1</td><td>99.0 78.8</td><td>99.3 80.7</td><td>76.5</td><td>99.3 82.0</td><td>99.2 73.8</td><td>99.1 80.8</td><td></td><td>99.9</td></tr><tr><td>51.0 95.6 66.1</td><td>36.1 53.1</td><td>34.1</td><td>47.8</td><td>44.2</td><td>56.4</td><td>79.5</td><td>25.9</td><td>85.3 69.5</td><td>83.3 88.9</td></tr></table>
459
+
460
+ number of true positives (tp) over the sum of true positives and false positives (fp), i.e.
461
+
462
+ $$
463
+ { \mathrm { p r e c i s i o n } } = { \frac { \mathrm { t p } } { \mathrm { t p } + \mathrm { f p } } } .
464
+ $$
465
+
466
+ Recall is defined as
467
+
468
+ $$
469
+ { \mathrm { r e c a l l } } = { \frac { \mathrm { t p } } { \mathrm { t p } + \mathrm { f n } } } ,
470
+ $$
471
+
472
+ where fn is the number of false negatives. We report the AUPR for all models and all datasets in Table 6. Qualitatively we find that the results do not differ from the ones reported in Table 2.
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1
+ # PDE-NET: LEARNING PDES FROM DATA
2
+
3
+ Anonymous authors Paper under double-blind review
4
+
5
+ # ABSTRACT
6
+
7
+ Partial differential equations (PDEs) play a prominent role in many disciplines such as applied mathematics, physics, chemistry, material science, computer science, etc. PDEs are commonly derived based on physical laws or empirical observations. However, the governing equations for many complex systems in modern applications are still not fully known. With the rapid development of sensors, computational power, and data storage in the past decade, huge quantities of data can be easily collected and efficiently stored. Such vast quantity of data offers new opportunities for data-driven discovery of hidden physical laws. Inspired by the latest development of neural network designs in deep learning, we propose a new feed-forward deep network, called PDE-Net, to fulfill two objectives at the same time: to accurately predict dynamics of complex systems and to uncover the underlying hidden PDE models. The basic idea of the proposed PDE-Net is to learn differential operators by learning convolution kernels (filters), and apply neural networks or other machine learning methods to approximate the unknown nonlinear responses. Comparing with existing approaches, which either assume the form of the nonlinear response is known or fix certain finite difference approximations of differential operators, our approach has the most flexibility by learning both differential operators and the nonlinear responses. A special feature of the proposed PDE-Net is that all filters are properly constrained, which enables us to easily identify the governing PDE models while still maintaining the expressive and predictive power of the network. These constrains are carefully designed by fully exploiting the relation between the orders of differential operators and the orders of sum rules of filters (an important concept originated from wavelet theory). We also discuss relations of the PDE-Net with some existing networks in computer vision such as Network-In-Network (NIN) and Residual Neural Network (ResNet). Numerical experiments show that the PDE-Net has the potential to uncover the hidden PDE of the observed dynamics, and predict the dynamical behavior for a relatively long time, even in a noisy environment.
8
+
9
+ # 1 INTRODUCTION
10
+
11
+ Differential equations, especially partial differential equations(PDEs), play a prominent role in many disciplines to describe the governing physical laws underlying a given system of interest. Traditionally, PDEs are derived based on simple physical principles such as conservation laws, minimum energy principles, or based on empirical observations. Important examples include the NavierStokes equations in fluid dynamics, the Maxwell’s equations for electromagnetic propagation, and the Schrodinger’s equations in quantum mechanics. However, many complex systems in modern ¨ applications (such as many problems in climate science, neuroscience, finance, etc.) still have eluded mechanisms, and the governing equations of these systems are only partially known. With the rapid development of sensors, computational power, and data storage in the last decade, huge quantities of data can be easily collected and efficiently stored . Such vast quantity of data offers new opportunities for data-driven discovery of potentially new physical laws. Then, one may ask the following interesting and intriguing question: can we learn a PDE model (if there exists one) from a given data set and perform accurate and efficient predictions using the learned model?
12
+
13
+ One of earlier attempts on data-driven discovery of hidden physical laws is by Bongard & Lipson (2007) and Schmidt & Lipson (2009). Their main idea is to compare numerical differentiations of the experimental data with analytic derivatives of candidate functions, and apply the symbolic regression and the evolutionary algorithm to determining the nonlinear dynamical system. Recently,
14
+
15
+ Brunton et al. (2016), Schaeffer (2017), Rudy et al. (2017) and Wu & Zhang (2017) propose an alternative approach using sparse regression. They construct a dictionary of simple functions and partial derivatives that are likely to appear in the unknown governing equations. Then, they take advantage of sparsity promoting techniques to select candidates that most accurately represent the data. When the form of the nonlinear response of a PDE is known, except for some scalar parameters, Raissi & Karniadakis (2017) presented a framework to learn these unknown parameters by introducing regularity between two consecutive time step using Gaussian process. More recently, Raissi et al. (2017) introduced a new class of universal function approximators called the physics informed neural networks which is capable of discovering nonlinear PDEs parameterized by scalars.
16
+
17
+ These recent work greatly advanced the progress of the problem. However, symbolic regression is expensive and does not scale very well to large systems. The sparse regression method requires to fix certain numerical approximations of the spatial differentiations in the dictionary beforehand, which limits the expressive and predictive power of the dictionary. Although the framework presented by Raissi & Karniadakis (2017); Raissi et al. (2017) is able to learn hidden physical laws using less data than the approach of sparse regression, the explicit form of the PDEs are assumed to be known except for a few scalar learnable parameters. Therefore, extracting governing equations from data in a less restrictive setting remains a great challenge.
18
+
19
+ The main objective of this paper is to accurately predict the dynamics of complex systems and to uncover the underlying hidden PDE models (should they exist) at the same time, with minimal prior knowledge on the systems. Our inspiration comes from the latest development of deep learning techniques in computer vision. An interesting fact is that some popular networks in computer vision, such as ResNet(He et al., 2016a;b), have close relationship with PDEs (Chen et al., 2015; E, 2017; Haber & Ruthotto, 2017; Sonoda & Murata, 2017; Lu et al., 2017). Furthermore, the deeper is the network, the more expressive power the network possesses, which may enable us to learn more complex dynamics arose from fields other than computer vision. However, existing deep networks designed in deep learning mostly emphasis on expressive power and prediction accuracy. These networks are not transparent enough to be able to reveal the underlying PDE models, although they may perfectly fit the observed data and perform accurate predictions. Therefore, we need to carefully design the network by combining knowledge from deep learning and applied mathematics so that we can learn the governing PDEs of the dynamics and make accurate predictions at the same time. Note that our work is closely related to Chen et al. (2015) where the authors designed their network based on discretization of quasilinear parabolic equations. However, it is not clear if the dynamics of image denoising has to be governed by PDEs, nor did the authors attempt to recover the PDE (should there exists one).
20
+
21
+ In this paper, we design a deep feed-forward network, named PDE-Net, based on the following generic nonlinear evolution PDE
22
+
23
+ $$
24
+ u _ { t } = F ( x , u , \nabla u , \nabla ^ { 2 } u , \ldots ) , \quad x \in \Omega \subset \mathbb { R } ^ { 2 } , \quad t \in [ 0 , T ] .
25
+ $$
26
+
27
+ The objective of the PDE-Net is to learn the form of the nonlinear response $F$ and to perform accurate predictions. Unlike the existing work, the proposed network only requires minor knowledge on the form of the nonlinear response function $F$ , and requires no knowledge on the involved differential operators (except for their maximum possible order) and their associated discrete approximations. The nonlinear response function $F$ can be learned using neural networks or other machine learning methods, while discrete approximations of the differential operators are learned using convolution kernels (i.e. filters) jointly with the learning of the response function $F$ . If we have a prior knowledge on the form of the response function $F$ , we can easily adjust the network architecture by taking advantage of the additional information. This may simplify the training and improve the results. We will also discuss relations of the PDE-Net to some existing networks in computer vision such as Network-In-Network (NIN) and ResNet. Details are given in Section 2.
28
+
29
+ In Section 3 and Section 4, we conduct numerical experiments on a linear PDE (convection-diffusion equation) and a nonlinear PDE (convection-diffusion equation with a nonlinear source). We generate data set for each PDE using high precision numerical methods and add Gaussian noise to mimic real situations. Our numerical results show that the PDE-Net can uncover the hidden equations of the observed dynamics, and can predict the dynamical behavior for a relatively long time, even in a noisy environment.
30
+
31
+ A particular novelty of our approach is that we impose appropriate constraints on the learnable filters in order to easily identify the governing PDE models while still maintaining the expressive and predictive power of the network. This makes our approach different from existing deep convolutional networks which mostly emphasis on the prediction accuracy of the networks, as well as all the existing approaches of learning PDEs from data which assume either the form of the response function is known or have fixed approximations of the differential operators. In other words, our proposed approach not only has vast flexibility in fitting observed dynamics and is able to accurately predict its future behavior, but is also able to reveal the hidden equations driving the observed dynamics. The constraints on the filters are motivated by the earlier work of Cai et al. (2012); Dong et al. (2017) where general relations between wavelet frame transforms and differential operators were established. In particular, it was observed in Dong et al. (2017) that we can relate filters and finite difference approximation of differential operators by examining the orders of sum rules of the filters (an important concept in wavelet theory and closely related to vanishing moments of wavelet functions). These constraints on the filters may also be useful in network designs for machine learning tasks in computer vision.
32
+
33
+ # 2 PDE-NET: A FLEXIBLE DEEP ARCHTECTURE TO LEARN PDES FROM DATA
34
+
35
+ Given a series of measurements of some physical quantities $\{ u ( t , \cdot ) : t = t _ { 0 } , t _ { 1 } , \cdot \cdot \cdot \}$ on the spatial domain $\Omega \subset \mathbb { R } ^ { 2 }$ , with $u ( t , \cdot ) : \Omega \mapsto \mathbb { R }$ , we want to discover the governing PDEs of the data. We assume that the observed data are associated with a PDE that takes the following general form:
36
+
37
+ $$
38
+ u _ { t } ( t , x , y ) = F ( x , y , u , u _ { x } , u _ { y } , u _ { x x } , u _ { x y } , u _ { y y } , \ldots ) , \quad ( x , y ) \in \Omega \subset \mathbb { R } ^ { 2 } , t \in [ 0 , T ] .
39
+ $$
40
+
41
+ Our objective is to design a feed-forward network, named the PDE-Net, that approximates the PDE (1) in the way that: 1) we can predict the dynamical behavior of the equation for as long time as possible; 2) we are able to reveal the form of the response function $F$ and the differential operators involved. There are two main components of the PDE-Net that are combined together in the same network: one is automatic determination on the differential operators involved in the PDE and their discrete approximations; the other is to approximate the nonlinear response function $F$ . In this section, we start with discussions on the relation between convolutions and differentiations in discrete setting.
42
+
43
+ # 2.1 CONVOLUTIONS AND DIFFERENTIATIONS
44
+
45
+ A comprehensive analysis on the relations between convolutions and differentiations within variational and PDE framework were laid out by Cai et al. (2012) and Dong et al. (2017), where the authors established general connections between PDE based approach and wavelet frame based approach for image restoration problems. We demonstrate one of the key observations of their work using a simple example. Consider the 2-dimensional Haar wavelet frame filter bank contains one low-pass filter $h _ { 0 0 }$ and three high pass filters $h _ { 1 0 } , h _ { 0 1 }$ and $h _ { 1 1 }$ :
46
+
47
+ $$
48
+ h _ { 0 0 } = \frac { 1 } { 4 } \left( \begin{array} { c c } { 1 } & { 1 } \\ { 1 } & { 1 } \end{array} \right) , h _ { 1 0 } = \frac { 1 } { 4 } \left( \begin{array} { c c } { 1 } & { - 1 } \\ { 1 } & { - 1 } \end{array} \right) , h _ { 0 1 } = \frac { 1 } { 4 } \left( \begin{array} { c c } { 1 } & { 1 } \\ { - 1 } & { - 1 } \end{array} \right) , h _ { 1 1 } = \frac { 1 } { 4 } \left( \begin{array} { c c } { 1 } & { - 1 } \\ { - 1 } & { 1 } \end{array} \right) .
49
+ $$
50
+
51
+ The associated Haar wavelet frame transform on an image $u$ is defined by
52
+
53
+ $$
54
+ W u = \{ h _ { i j } [ - \cdot ] \circledast u : 0 \leq i , j \leq 1 \} ,
55
+ $$
56
+
57
+ where $\circledast$ is the circular convolution. It is easy to verify using Taylor’s expansion that the high frequency coefficients of the Haar wavelet frame transform on $u$ are discrete approximations of differential operators:
58
+
59
+ $$
60
+ h _ { 1 0 } [ - \cdot ] \circledast u \approx \frac { 1 } { 2 } \delta _ { x } u _ { x } , \ h _ { 0 1 } [ - \cdot ] \circledast u \approx \frac { 1 } { 2 } \delta _ { y } u _ { y } , \ h _ { 1 1 } [ - \cdot ] \circledast u \approx \frac { 1 } { 4 } \delta _ { x } \delta _ { y } u _ { x y } .
61
+ $$
62
+
63
+ Here, $\delta _ { x }$ and $\delta _ { y }$ represent the horizontal and vertical spatial grid size respectively. For simplicity of notation, we use regular character to denote both discrete and continuum functions, since there should be no confusion within the context.
64
+
65
+ A profound relationship between convolutions and differentiations was presented in Dong et al. (2017), where the authors discussed the connection between the order of sum rules of filters and the orders of differential operators. Note that the order of sum rules is closely related to the order of vanishing moments in wavelet theory (Daubechies, 1992; Mallat, 1999). We first recall the definition of the order of sum rules.
66
+
67
+ Definition 2.1 (Order of Sum Rules). For a filter $q$ , we say $q$ to have sum rules of order $\alpha =$ $\left( \alpha _ { 1 } , \alpha _ { 2 } \right)$ , where $\alpha \in \mathbb { Z } _ { + } ^ { 2 }$ , provided that
68
+
69
+ $$
70
+ \sum _ { k \in \mathbb { Z } ^ { 2 } } k ^ { \beta } q [ k ] = 0
71
+ $$
72
+
73
+ for all $\beta \in \mathbb { Z } _ { + } ^ { 2 }$ with $| \beta | < | \alpha |$ and for all $\beta \in \mathbb { Z } _ { + } ^ { 2 }$ with $| \beta | = | \alpha |$ but $\beta \neq \alpha$ . If (2) holds for all $\beta \in \mathbb { Z } _ { + } ^ { 2 }$ with $| \beta | < K$ except for $\beta \neq \beta _ { 0 }$ with certain $\beta _ { 0 } \in \mathbb { Z } _ { + } ^ { 2 }$ and $\left. \beta _ { 0 } \right. = J < K$ , then we say q to have total sum rules of order $K \backslash \{ J + 1 \}$ .
74
+
75
+ The following proposition from Dong et al. (2017) links the orders of sum rules with orders of differential operator.
76
+
77
+ Propositin 2.1. Let q be a filter with sum rules of order $\alpha \in \mathbb { Z } _ { + } ^ { 2 }$ . Then for a smooth function $F ( x )$ on $\mathbb { R } ^ { 2 }$ , we have
78
+
79
+ $$
80
+ { \frac { 1 } { \varepsilon ^ { | \alpha | } } } \sum _ { k \in \mathbb { Z } ^ { 2 } } q [ k ] F ( x + \varepsilon k ) = C _ { \alpha } { \frac { \partial ^ { \alpha } } { \partial x ^ { \alpha } } } F ( x ) + O ( \varepsilon ) , a s \varepsilon \to 0 ,
81
+ $$
82
+
83
+ where $C _ { \alpha }$ is the constant defined by
84
+
85
+ $$
86
+ C _ { \alpha } = \frac { 1 } { \alpha ! } \sum _ { k \in \mathbb { Z } ^ { 2 } } k ^ { \alpha } q [ k ] .
87
+ $$
88
+
89
+ If, in addition, $q$ has total sum rules of order $K \backslash \{ | \alpha | + 1 \}$ for some $K > | \alpha |$ , then
90
+
91
+ $$
92
+ { \frac { 1 } { \varepsilon ^ { | \alpha | } } } \sum _ { k \in \mathbb { Z } ^ { 2 } } q [ k ] F ( x + \varepsilon k ) = C _ { \alpha } { \frac { \partial ^ { \alpha } } { \partial x ^ { \alpha } } } F ( x ) + O ( \varepsilon ^ { K - | \alpha | } ) , a s \varepsilon \to 0 .
93
+ $$
94
+
95
+ According to Proposition 2.1, an $\alpha$ th order differential operator can be approximated by the convolution of a filter with $\alpha$ order of sum rules. Furthermore, according to (4), one can obtain a high order approximation of a given differential operator if the corresponding filter has an order of total sum rules with $K > | \alpha | + k , k \geqslant 1$ . For example, the filter $h _ { 1 0 }$ in the Haar wavelet frame filter bank has a sum rules of order $( 1 , 0 )$ , and a total sum rules of order $2 \backslash \{ 2 \}$ . Thus, up to a constant and a proper scaling, h10 corresponds to a discretization of ∂∂x with first order approximation. The filer $h _ { 1 1 }$ has a sum rules of order $( 1 , 1 )$ , and a total sum rules of order $3 \backslash \{ 3 \}$ . Thus, up to a constant and a proper scaling, $h _ { 1 1 }$ corresponds to a discretization of $\frac { \partial ^ { 2 } } { \partial x \partial y }$ with first order approximation. Finally, consider filter
96
+
97
+ $$
98
+ q = \left( \begin{array} { c c c } { { 1 } } & { { 0 } } & { { - 1 } } \\ { { 2 } } & { { 0 } } & { { - 2 } } \\ { { 1 } } & { { 0 } } & { { - 1 } } \end{array} \right) .
99
+ $$
100
+
101
+ It has a sum rules of order $( 1 , 0 )$ , and a total sum rules of order $3 \backslash \{ 2 \}$ . Thus, up to a constant and a proper scaling, q corresponds to a discretization of ∂∂x with second order approximation.
102
+
103
+ Now, we introduce the concept of moment matrix for a given filter that will be used to constrain filters in the PDE-Net. For an $N \times N$ filter $q$ , define the moment matrix of $q$ as
104
+
105
+ $$
106
+ M ( q ) = ( m _ { i , j } ) _ { N \times N } , \mathrm { ~ w h e r e ~ } m _ { i , j } = \frac { 1 } { ( i - 1 ) ! ( j - 1 ) ! } \sum _ { k \in \mathbb { Z } ^ { 2 } } k _ { 1 } ^ { i - 1 } k _ { 2 } ^ { j - 1 } q [ k _ { 1 } , k _ { 2 } ] ,
107
+ $$
108
+
109
+ for $i , j = 1 , 2 , \dots , N$ . We shall call the $( i , j )$ -element of $M ( q )$ the $( i - 1 , j - 1 )$ -moment of $q$ for simplicity. Combining (5) and Proposition 2.1, one can easily see that filter $q$ can be designed to approximate any differential operator at any given approximation order by imposing constraints on $M ( q )$ . For example, if we want to approximate $\textstyle { \frac { \partial u } { \partial x } }$ (up to a constant) by convolution $q \circledast u$ where $q$ is a $3 \times 3$ filter, we can consider the following constrains on $M ( q )$ :
110
+
111
+ $$
112
+ \left( \begin{array} { c c c } { { 0 } } & { { 0 } } & { { \star } } \\ { { 1 } } & { { \star } } & { { \star } } \\ { { \star } } & { { \star } } & { { \star } } \end{array} \right) \quad \mathrm { o r } \quad \left( \begin{array} { c c c } { { 0 } } & { { 0 } } & { { 0 } } \\ { { 1 } } & { { 0 } } & { { \star } } \\ { { 0 } } & { { \star } } & { { \star } } \end{array} \right) .
113
+ $$
114
+
115
+ Here, $\star$ means no constraint on the corresponding entry. The constraints described by the moment matrix on the left of (6) guarantee the approximation accuracy is at least first order, and the ones on
116
+
117
+ the right guarantee an approximation of at least second order. In particular, when all entries of $M ( q )$ are constrained, e.g.
118
+
119
+ $$
120
+ M ( q ) = \left( \begin{array} { c c c } { { 0 } } & { { 0 } } & { { 0 } } \\ { { 1 } } & { { 0 } } & { { 0 } } \\ { { 0 } } & { { 0 } } & { { 0 } } \end{array} \right) ,
121
+ $$
122
+
123
+ the corresponding filter can be uniquely determined, in which case we call it a “frozen” filter. In the PDE-Net which shall be introduced in the next subsection, all filters are learned subjected to partial constraints on their associated moment matrices.
124
+
125
+ It is worth noticing that the approximation property of a filter is limited by its size. Generally speaking, large filters can approximate higher order differential operators or lower order differential operators with higher approximation orders. Taking 1-dimensional case as an example, 3-element filters cannot approximate the fifth order differential operator, whereas 7-element filters can. In other words, the larger are the filters, the stronger is the representation capability of filters. However, larger filters lead to more memory overhead and higher computation cost. It is a wisdom to balance the trade-off in practice.
126
+
127
+ # 2.2 ARCHITECTURE OF PDE-NET
128
+
129
+ Given the evolution PDE (1), we consider forward Euler as the temporal discretization. One may consider more sophisticated temporal discretization which leads to different network architectures. For simplicity, we focus on forward Euler in this paper.
130
+
131
+ # ONE $\delta t$ -BLOCK:
132
+
133
+ Let $\tilde { u } ( t _ { i + 1 } , \cdot )$ be the predicted value of $u$ at time $t _ { i + 1 }$ based on the value of $u$ at $t _ { i }$ . Then, we have
134
+
135
+ $$
136
+ \tilde { u } ( t _ { i + 1 } , \cdot ) = D _ { 0 } u ( t _ { i } , \cdot ) + \Delta t \cdot F ( x , y , D _ { 0 0 } u , D _ { 1 0 } u , D _ { 0 1 } u , D _ { 2 0 } u , D _ { 1 1 } u , D _ { 0 2 } u , \dots ) .
137
+ $$
138
+
139
+ Here, the operators $D _ { 0 }$ and $D _ { i j }$ are convolution operators with the underlying filters denoted by $q _ { 0 }$ and $q _ { i j }$ , i.e. $D _ { 0 } u = q _ { 0 } \circledast u$ and $D _ { i j } u = q _ { i j } \circledast u$ . The operators $D _ { 1 0 }$ , $D _ { 0 1 }$ , $D _ { 1 1 }$ , etc. approximate differential operators, i.e. $\begin{array} { r } { D _ { i j } u \approx \frac { \partial ^ { i + j } u } { \partial ^ { i } x \partial ^ { j } y } } \end{array}$ . The operators $D _ { 0 }$ and $D _ { 0 0 }$ are average operators. The purpose of introducing these average operators in stead of using the identity is to improve stability of the network and enables it to capture more complex dynamics. Other than the assumption that the observed dynamics is governed by a PDE of the form (1), we assume that the highest order of the PDE is less than some positive integer. Then, the task of approximating $F$ is equivalent to a multivariate regression problem, which can be approximated by a point-wise neural network (with shared weights across the computation domain $\Omega$ ) or other classical machine learning methods. Combining the approximation of differential operators and the nonlinear function $F$ , we achieve an approximation framework of (7) which will be referred to as a $\delta t$ -block (see Figure 1). Note that if we have a prior knowledge on the form of the response function $F$ , we can easily adjust the network architecture by taking advantage of the additional information. This may simplify the training and improve the results.
140
+
141
+ # PDE-NET (MULTIPLE $\delta t$ -BLOCKS):
142
+
143
+ One $\delta t$ -block only guarantees the accuracy of one-step dynamics, which does not take error accumulation into consideration. This may cause severe instability in prediction. To improve the stability of the network and enable long-term prediction, we stack multiple $\delta t$ -blocks into a deep network, and call this network the $P D E$ -Net (see Figure 2). The importance of stacking multiple $\delta t$ -blocks will be demonstrated in Section 3.
144
+
145
+ The PDE-Net can be easily described as: (1) stacking one $\delta t$ -block multiple times; (2) sharing parameters in all $\delta t$ -blocks. Given an input data $u ( t _ { i } , \cdot )$ , training a PDE-Net with n $\delta t$ -blocks needs to minimize the accumulated error $| | u ( t _ { i + n } , \cdot ) - \tilde { u } ( t _ { i + n } , \cdot ) | | _ { 2 } ^ { 2 }$ , where $\tilde { u } ( t _ { i + n } , \cdot )$ is the output from the PDE-Net (i.e. n $\delta t$ -blocks) with input $u ( t _ { i } , \cdot )$ . Thus, the PDE-Net with bigger $n$ owns a longer time stability. Note that sharing parameters is a common practice in deep learning, which decreases the number of parameters and leads to significant memory reduction (Goodfellow et al., 2016).
146
+
147
+ ![](images/bc297c2ec66e617b790aada20ecc98f5f4c81af4e9d1be10869a44e90a2dfa78.jpg)
148
+ Figure 1: The schematic diagram of a $\delta t$ -block.
149
+
150
+ ![](images/5f29b52923eefbb55f812820527e47c45f872de3a0be0cd1e9060bf1c49aaf25.jpg)
151
+ Figure 2: The schematic diagram of the PDE-Net: multiple $\delta t$ -blocks.
152
+
153
+ # LOSS FUNCTION AND CONSTRAINTS:
154
+
155
+ Consider the data set $\{ u _ { j } ( t _ { i } , \cdot ) : i , j = 0 , 1 , \ldots \}$ , where $j$ indicates the $j$ -th solution path with a certain initial condition of the unknown dynamics. We would like to train the PDE-Net with $n$ $\delta t$ -blocks. For a given $n \geq 1$ , every pair of the data $\{ u _ { j } ( t _ { i } , \cdot ) , u _ { j } ( t _ { i + n } , \cdot ) \}$ , for each $i$ and $j$ , is a training sample, where $u _ { j } ( t _ { i } , \cdot )$ is the input and $u _ { j } \big ( t _ { i + n } , \cdot \big )$ is the label that we need to match with the output from the PDE-Net. We select the following simple $\ell _ { 2 }$ loss function for training:
156
+
157
+ $$
158
+ L = \sum _ { i , j } l _ { i j } , \mathrm { w h e r e } \quad l _ { i j } = | | u _ { j } ( t _ { i + n } , \cdot ) - \tilde { u } _ { j } ( t _ { i + n } , \cdot ) | | _ { 2 } ^ { 2 } ,
159
+ $$
160
+
161
+ where $\tilde { u } _ { j } \big ( t _ { i + n } , \cdot \big )$ is the output of the PDE-Net with $u _ { j } ( t _ { i } , \cdot )$ as the input.
162
+
163
+ All the filters involved in the PDE-Net are properly constrained using their associated moment matrices. Let $q _ { 0 }$ and $q _ { i j }$ be the underlying filters of $D _ { 0 }$ and $D _ { i j }$ . We impose the following constrains
164
+
165
+ $$
166
+ ( M ( q _ { 0 } ) ) _ { 1 , 1 } = 1 , \quad ( M ( q _ { 0 0 } ) ) _ { 1 , 1 } = 1
167
+ $$
168
+
169
+ and for $i + j > 0$
170
+
171
+ $$
172
+ \left\{ \begin{array} { l l } { ( M ( q _ { i , j } ) ) _ { k _ { 1 } , k _ { 2 } } = 0 } & { k _ { 1 } + k _ { 2 } \leq i + j + 2 , ( k _ { 1 } , k _ { 2 } ) \neq ( i + 1 , j + 1 ) , } \\ { ( M ( q _ { i , j } ) ) _ { k _ { 1 } , k _ { 2 } } = 1 } & { ( k _ { 1 } , k _ { 2 } ) = ( i + 1 , j + 1 ) . } \end{array} \right.
173
+ $$
174
+
175
+ For example, for $3 \times 3$ filters, we have
176
+
177
+ $$
178
+ M ( q _ { 0 } ) = M ( q _ { 0 0 } ) = \left( \begin{array} { c c c } { { 1 } } & { { \star } } & { { \star } } \\ { { \star } } & { { \star } } & { { \star } } \\ { { \star } } & { { \star } } & { { \star } } \end{array} \right)
179
+ $$
180
+
181
+ and
182
+
183
+ $$
184
+ M ( q _ { 1 0 } ) = \left( \begin{array} { c c c } { { 0 } } & { { 0 } } & { { \star } } \\ { { 1 } } & { { \star } } & { { \star } } \\ { { \star } } & { { \star } } & { { \star } } \end{array} \right) , M ( q _ { 0 1 } ) = \left( \begin{array} { c c c } { { 0 } } & { { 1 } } & { { \star } } \\ { { 0 } } & { { \star } } & { { \star } } \\ { { \star } } & { { \star } } & { { \star } } \end{array} \right) , M ( q _ { 1 1 } ) = \left( \begin{array} { c c c } { { 0 } } & { { 0 } } & { { 0 } } \\ { { 0 } } & { { 1 } } & { { \star } } \\ { { 0 } } & { { \star } } & { { \star } } \end{array} \right) , \dots .
185
+ $$
186
+
187
+ To demonstrate the necessity of learnable filters, we will compare the PDE-Net having the aforementioned constrains on the filters with the PDE-Net having frozen filters. To differentiate the two cases, we shall call the PDE-Net with frozen filters “the Frozen-PDE-Net”.
188
+
189
+ To further increase the expressive power and flexibility of the PDE-Net, we may associate multiple filters to approximate a given differential operator. However, in order not to mess up the identifiability of the underlying PDE model, we may select only one of the filters to provide correct approximation to the given differential operator in the way as described above. The rest of the filters are constrained in the way that they only contribute to modify the local truncation errors. For example, consider two $3 \times 3$ filters $\{ q _ { 0 } , q _ { 1 } \}$ and constrain their moment matrices as follows
190
+
191
+ $$
192
+ M ( q _ { 0 } ) = \left( \begin{array} { c c c } { { 0 } } & { { 0 } } & { { \star } } \\ { { 1 } } & { { \star } } & { { \star } } \\ { { \star } } & { { \star } } & { { \star } } \end{array} \right) , M ( q _ { 1 } ) = \left( \begin{array} { c c c } { { 0 } } & { { 0 } } & { { \star } } \\ { { 0 } } & { { \star } } & { { \star } } \\ { { \star } } & { { \star } } & { { \star } } \end{array} \right) .
193
+ $$
194
+
195
+ Then, $q _ { 0 } \circledast u + q _ { 1 } \circledast u$ is potentially a better approximation to $u _ { x }$ (up to a constant) than $q _ { 0 } \circledast u$ . However, for simplicity, we only use one filter to approximate a given differential operator in this paper.
196
+
197
+ # NOVELTY OF THE PDE-NET:
198
+
199
+ Different from fixing numerical approximations of differentiations in advance in sparse regression methods (Schaeffer, 2017; Rudy et al., 2017), using learnable filters makes the PDE-Net more flexible, and enables more robust approximation of unknown dynamics and longer time prediction (see numerical experiments in Section 3 and Section 4). Furthermore, the specific form of the response function $F$ is also approximated from the data, rather than assumed to be known in advance (such as (Raissi & Karniadakis, 2017)). On the other hand, by inflicting constrains on moment matrices, we can identify which differential operators are included in the underlying PDE which helps with identifying the nonlinear response function $F$ . This grants transparency to the PDE-Net and the potential to reveal hidden physical laws. Therefore, the proposed PDE-Net is distinct from the existing learning based method to discover PDEs from data, as well as networks designed in deep learning for computer vision tasks.
200
+
201
+ # 2.3 INITIALIZATION AND TRAINING
202
+
203
+ In the PDE-Net, parameters can be divided into three groups:
204
+
205
+ • filters to approximate differential operators;
206
+ • the parameters of the point-wise neural network to approximate $F$ ;
207
+ • hyper-parameters, such as the number of filters, the size of filters, the number of layers, etc.
208
+
209
+ The parameters of the point-wise neural network are shared across the computation domain $\Omega$ , and are initialized by random sampling from a Gaussian distribution. For the filters, we initialize them by freezing them to their corresponding differential operators. For example, if a filter is to approximate $\frac { \partial } { \partial x }$ , we freeze it by constraining its $( 1 , 0 )$ -moment to 1 and other moments to 0. During the training process, we release the filters by switching to the constrains described in Section 2.2.
210
+
211
+ Instead of training an $n$ -layer PDE-Net directly, we adopt layer-wise training, which improves the training speed. To be more precise, we start with training the PDE-Net on the first $\delta t$ -block, and then use the results of the first $\delta t$ -block as the initialization and restart training the PDE-Net on the first two $\delta t$ -blocks. Repeat until we complete all $n$ blocks. Note that all the parameters in each of the $\delta t$ -block are shared across layers. In addition, we add a warm-up step before the training of the first $\delta t$ -block. The warm-up step is to obtain a good initial guess of the parameters of the point-wise neural network that approximates $F$ by using frozen filters.
212
+
213
+ # 2.4 RELATIONS TO SOME EXISTING NETWORKS
214
+
215
+ In recent years, a variety of deep neural networks have been introduced with great success in computer vision. The structure of the proposed PDE-Net is similar to some existing networks such as the Network-In-Network (NIN) (Lin et al., 2013) and the deep Residual Neural Network (ResNet) (He et al., 2016a;b).
216
+
217
+ The NIN is an improvement over the traditional convolutional neural networks. One of the special designs of NIN is the use of multilayer perceptron convolution (mlpconv) layers instead of the ordinary convolution layers. An mlpconv layer contains the convolutions and small point-wise neural networks. Such design can improve the ability of the network to extract nonlinear features from shallow layers. The inner structure of one $\delta t$ -block of the PDE-Net is similar to the mlpconv layer, and the multiple $\delta t$ -blocks structure is similar to the NIN structure, except for the pooling and ReLU operations.
218
+
219
+ On the other hand, each $\delta t$ -block of the PDE-Net has two paths (see Figure 1 and Figure 2): one is for the averaged quantity of $u$ and the other is for the increment $F$ . This structure coincides with the “residual block” introduced in ResNet. In fact, there has been a substantial study on the relation between ResNet and dynamical systems recently (E, 2017; Haber & Ruthotto, 2017; Sonoda & Murata, 2017).
220
+
221
+ # 3 NUMERICAL STUDIES: CONVECTION-DIFFUSION EQUATIONS
222
+
223
+ Convection-diffusion equations are classical PDEs that are used to describe physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection (Chandrasekhar, 1943). Convection-diffusion equations are widely applied in many scientific areas and industrial fields, such as pollutants dispersion in rivers or atmosphere, solute transferring in a porous medium, and oil reservoir simulation. In practical situations, usually the physical and chemical properties on different locations cannot be the same (called anisotropy in physics), thus it is more reasonable that convection coefficients and diffusion coefficients are variables instead of constants.
224
+
225
+ # 3.1 SIMULATED DATA, TRAINING AND TESTING
226
+
227
+ We consider a 2-dimensional linear variable-coefficient convection-diffusion equation on $\Omega \ =$ $[ 0 , 2 \pi ] \times [ 0 , 2 \pi ]$ ,
228
+
229
+ $$
230
+ \left\{ \begin{array} { l l } { \frac { \partial u } { \partial t } } & { = a ( x , y ) u _ { x } + b ( x , y ) u _ { y } + c u _ { x x } + d u _ { y y } \qquad \mathrm { w i t h ~ } ( t , x , y ) \in [ 0 , 0 . 2 ] \times \Omega , } \\ { u | _ { t = 0 } } & { = u _ { 0 } ( x , y ) , } \end{array} \right.
231
+ $$
232
+
233
+ where
234
+
235
+ $$
236
+ a ( x , y ) = 0 . 5 ( \cos ( y ) + x ( 2 \pi - x ) \sin ( x ) ) + 0 . 6 , b ( x , y ) = 2 ( \cos ( y ) + \sin ( x ) ) + 0 . 8 ,
237
+ $$
238
+
239
+ $c = 0 . 2$ and $d = 0 . 3$
240
+
241
+ The computation domain $\Omega$ is discretized using a $5 0 \times 5 0$ regular mesh. Data is generated by solving problem (8) using a high precision numerical scheme with pseudo-spectral method for spatial discretization and 4th order Runge-Kutta for temporal discretization (with time step size $\delta t = 0 . 0 1$ ). We assume periodic boundary condition and the initial value $u _ { 0 } ( x , y )$ is generated from
242
+
243
+ $$
244
+ u _ { 0 } ( x , y ) = \sum _ { | k | , | l | \leq N } \lambda _ { k , l } \cos ( k x + l y ) + \gamma _ { k , l } \sin ( k x + l y ) ,
245
+ $$
246
+
247
+ where $\begin{array} { r } { N = 9 , \lambda _ { k , l } , \gamma _ { k , l } \sim \mathcal { N } ( 0 , \frac { 1 } { 5 0 } ) } \end{array}$ , and $k$ and $l$ are chosen randomly. In order to mimic real world scenarios, we add noise to the generated data. For each sample sequence $u ( x , y , t ) , t \in [ 0 , 0 . 2 ]$ , the noise is added as
248
+
249
+ $$
250
+ \widehat { u } ( x , y , t ) = u ( x , y , t ) + 0 . 0 1 \times M W
251
+ $$
252
+
253
+ where $M = \operatorname* { m a x } _ { x , y , t } \{ u ( x , y , t ) \}$ , $W \sim \mathcal { N } ( 0 , 1 )$ and $\mathcal { N } ( 0 , 1 )$ represents the standard normal distribution.
254
+
255
+ Suppose we know a priori that the underlying PDE is linear with order no more than 4. Then, the response function $F$ takes the following form
256
+
257
+ $$
258
+ F = \sum _ { 0 \leq i + j \leq 4 } f _ { i j } ( x , y ) \frac { \partial ^ { i + j } u } { \partial x ^ { i } \partial y ^ { j } } .
259
+ $$
260
+
261
+ Each $\delta t$ -block of the PDE-Net can be written as
262
+
263
+ $$
264
+ \tilde { u } ( t _ { n + 1 } , \cdot ) = D _ { 0 } u ( t _ { n } , \cdot ) + \delta t \cdot ( c _ { 0 0 } D _ { 0 0 } u + c _ { 1 0 } D _ { 1 0 } u + \ldots + c _ { 0 4 } D _ { 0 4 } u ) ,
265
+ $$
266
+
267
+ where $\{ D _ { 0 } , D _ { i j } : i + j \le 4 \}$ are convolution operators and $\{ c _ { i j } : i + j \leq 4 \}$ are 2D arrays which approximate functions $f _ { i j } ( x , y )$ on $\Omega$ . The approximation is achieved using piecewise quadratic polynomial interpolation with smooth transitions at the boundaries of each piece. The filters associated to the convolution operators $\{ D _ { 0 } , D _ { i j } : i + j \le 4 \}$ and the coefficients of the piecewise quadratic polynomials are the trainable parameters of the network.
268
+
269
+ During training and testing, the data is generated on-the-fly, i.e. we only generate the data needed following the aforementioned procedure when training and testing the PDE-Net. In our experiments, the size of the filters that will be used is $5 \times 5$ or $7 \times 7$ . The total number of trainable parameters for each $\delta t$ -block is approximately $1 7 \mathbf { k }$ . During training, we use LBFGS, instead of SGD, to optimize the parameters. We use 28 data samples per batch to train each layer (i.e. $\delta t$ -block) and we only construct the PDE-Net up to 20 layers, which requires totally 560 data samples per batch. Note that the PDE-Net is designed with the assumption that it approximates nonlinear evolution PDEs, which is a relatively stronger assumption than the networks in deep learning. Therefore, we require less training data and LBFGS performs better than SGD (which is widely adopted in deep learning). Furthermore, as will be shown by our numerical results, the learned PDE-Net generalizes very well. The PDE-Net can accurately predict the dynamics even when the initial data $u _ { 0 }$ does not come from the same distribution as in the training process.
270
+
271
+ # 3.2 RESULTS AND DISCUSSIONS
272
+
273
+ This section presents numerical results of training the PDE-Net using the data set described in the previous subsection. We will specifically observe how the learned PDE-Net performs in terms of prediction of dynamical behavior and identification of the underlying PDE model. Furthermore, we will investigate the effects of some of the hyper-parameters (e.g. size of the filters, number of $\delta t$ -blocks) on the learned PDE-Net.
274
+
275
+ # PREDICTING LONG-TIME DYNAMICS
276
+
277
+ We demonstrate the ability of the trained PDE-Net in prediction, which in the language of machine learning is the ability to generalize. After the PDE-Net with $n \delta t$ -blocks $1 \leq n \leq 2 0 ,$ ) is trained, we randomly generate 560 initial guesses based on (9) and (10), feed them to the PDE-Net, and measure the normalized error between the predicted dynamics (i.e. the output of the PDE-Net) and the actual dynamics (obtained by solving (8) using high precision numerical scheme). The normalized error between the true data $u$ and the predicted data $\tilde { u }$ is defined as
278
+
279
+ $$
280
+ \epsilon = \frac { \| \tilde { u } - u \| _ { 2 } ^ { 2 } } { \| u - \bar { u } \| _ { 2 } ^ { 2 } } ,
281
+ $$
282
+
283
+ where $\bar { u }$ is the spatial average of $u$ . The error plots are shown in Figure 3. Results of longer prediction for the PDE-Net with $7 \times 7$ learnable filters are shown in Figure 4. Some of the images of the predicted dynamics are presented in Figure 5. From these results, we can see that:
284
+
285
+ • Even trained with noisy data, the PDE-Net is able to perform long-term prediction (see Figure 5);
286
+ • Having multiple $\delta t$ -blocks helps with the stability of the PDE-Net and ensures long-term prediction (see Figure 3);
287
+ • The PDE-Net performs significantly better than Frozen-PDE-Net, especially for $7 \times 7$ filters (see Figure 3);
288
+
289
+ • The PDE-Net with $7 \times 7$ filters significantly outperforms the PDE-Net with $5 \times 5$ filters in terms of the length of reliable predictions (see Figure 3 and 4). To reach an $O ( 1 )$ error, the length of prediction for the PDE-Net with $7 \times 7$ filters is about 10 times of that for the PDE-Net with $5 \times 5$ filters.
290
+
291
+ ![](images/a86b0e8b8b90fff95afaaaf51532d45ad8b4ad26686a3900cfe6b03acf525cc5.jpg)
292
+ Figure 3: Prediction errors of the PDE-Net (orange) and Frozen-PDE-Net (blue) with $5 \times 5$ (first row) and $7 \times 7$ (second row) filters. In each plot, the horizontal axis indicates the time of prediction in the interval $( 0 , 6 0 \times \delta t ] = ( 0 , 0 . 6 ]$ , and the vertical axis shows the normalized errors. The banded curves indicate the $2 5 \%$ & $7 5 \%$ percentile of the normalized errors among 560 test samples.
293
+
294
+ ![](images/8f3353e1f22d777d3572922c3187448ecc7608e98c988ca7ef70d091934e89f0.jpg)
295
+ Figure 4: Long-time prediction for the PDE-Net with $7 \times 7$ filters. The horizontal axis ranges in $( 0 , 5 ]$ . Time step $\delta t = 0 . 0 1$ .
296
+
297
+ ![](images/0e54673f5a1683ca4bdfdd8484a4e6ba50e6bdfd88bceefcb8c62f12cce00e9a.jpg)
298
+ Figure 5: Images of the true dynamics and the predicted dynamics. The first row shows the images of the true dynamics. The second row shows the images of the predicted dynamics using the PDE-Net having $3 \delta t$ -blocks with $5 \times 5$ and $7 \times 7$ filters. Time step $\delta t = 0 . 0 1$ .
299
+
300
+ # DISCOVERING THE HIDDEN EQUATION
301
+
302
+ For the linear problem, identifying the PDE amounts to finding the coefficients $\{ c _ { i j } : i + j \leq 4 \}$ that approximate $\{ f _ { i j } : i + j \stackrel { . } { \le } 4 \bar \}$ . The coefficients $\{ c _ { i j } : i + j \leq 2 \}$ of the trained PDE-Net are shown in Figure 6. Note that $\{ f _ { 1 1 } \} \cup \{ f _ { i j } : 2 < i + j \leq 4 \}$ are absent from the PDE (8), and the corresponding coefficients learned by the PDE-Net are indeed close to zero. In order to have a more concise demonstration of the results, we only show the image of $\{ c _ { i j } : i + j \leq 2 \}$ in Figure 6.
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+
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+ Comparing the first three rows of Figure 6, the coefficients $\{ c _ { i j } \}$ learned by the PDE-Net are close to the true coefficients $\{ f _ { i j } \}$ except for some oscillations due to the presence of noise in the training data. Furthermore, the last row of Figure 6 indicates that having multiple $\delta t$ -blocks helps with estimation of the coefficients. However, having larger filters does not seem to improve the learning of the coefficients, though it helps tremendously in prolonging predictions of the PDE-Net.
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+
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+ ![](images/324c09133edf71f24a8a615ea9f303813e20f3c2ff3fa2cebe780a0ecfc8b1d7.jpg)
307
+ Figure 6: First row: the true coefficients of the equation. From the left to right are coefficients of $u$ , $u _ { x }$ , $u _ { y }$ , $u _ { x x }$ , $u _ { x y }$ and $u _ { y y }$ . Second row: the learned coefficients by the PDE-Net with $6 \delta t$ -blocks and $5 \times 5$ filters. Third row: the learned coefficients by the PDE-Net with 6 $\delta t$ -blocks and $7 \times 7$ filters. Last row: the errors between true and learned coefficients v.s. number of $\delta t$ -blocks $( 1 , 2 , \ldots , 1 3 )$ with different sizes of filters (blue for $5 \times 5$ and orange for $7 \times 7$ ).
308
+
309
+ # FURTHER EXPERIMENTS
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+
311
+ To further demonstrate how well the learned PDE-Net generalizes, we generate initial values following (9) with highest frequency equal to 12, followed by adding noise (10). Note that the maximum allowable frequency in the training set is 9. The results of long-time prediction and the estimated dynamics are shown in Figure 7. Although oscillations are observed in the prediction, the estimated dynamic still captures the main pattern of the true dynamic.
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+
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+ The PDE (8) is of second order. In our previous experiments, we assumed that the PDE does not exceed the 4th order. If we know that the PDE is of second order, we will be able to have a more accurate estimation of the variable coefficients of the convection and diffusion terms. However, the prediction errors are slightly higher since we have fewer trainable parameters. Nonetheless, since we are using a more accurate prior knowledge on the unknown PDE, the variance of the prediction errors are smaller than before. These results are summarized in Figure 8 (green curves) and Figure 9.
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+
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+ To further demonstrate the importance of the moment constraints on the filters in the PDE-Net, we trained the network without any moment constraints and skipped any steps that utilize the knowledge of the relation between the filters and differential operators (i.e. we skipped warm-up and the initialization using finite difference filters). For simplicity, we call the PDE-Net train in this way as the Freed-PDE-Net. The prediction errors of the Freed-PDE-Net are shown as the red curves in Figure 8. Since without moment constraints, we do not know the correspondence of the filters with differential operators. Therefore, we cannot identify the correspondence of the learned variable coefficients either. We plot all the 15 variable coefficients (assuming the underlying PDE is of order $\leq 4$ ) in Figure 10. As one can see that the Freed-PDE-Net is better in prediction than the PDE-Net since it has more trainable parameters than the PDE-Net. However, we are unable to identify the PDE from the Free-PDE-Net.
316
+
317
+ ![](images/21d1a51e285a9d940726b4e56ee8dd53752c2d9791e620ee413b8bbe0d3bc053.jpg)
318
+ Figure 7: Testing with higher frequency initializations (linear convection-diffusion equation). First row: long-time prediction. Second row: estimated dynamics. Here, $\delta t = 0 . 0 1$ .
319
+
320
+ ![](images/c0086f94cdc977640ca30b698620529989c47e690128ded26a0de10aecfb26da.jpg)
321
+ Figure 8: Prediction errors of the PDE-Net assuming the underlying PDE has order $\leq 4$ (orange), order $\leq 2$ (green) and Freed-PDE-Net (red) with $7 \times 7$ filters. In each plot, the horizontal axis indicates the time of prediction in the interval $( 0 , 8 0 \times \delta t ] = ( 0 , 0 . 8 ]$ , and the vertical axis shows the normalized errors. The banded curves indicate the $2 5 \%$ & $7 5 \%$ percentile of the normalized errors among 560 test samples.
322
+
323
+ # QUICK SUMMARY:
324
+
325
+ In summary, the numerical experiments show that the PDE-Net is able to conduct accurate prediction and identify the underlying PDE model at the same time, even in a noisy environment. Multiple $\delta t$ -blocks, i.e. deeper structure of the PDE-Net, makes the PDE-Net more stable and enables longer time prediction. Furthermore, using larger filters helps with stability and can prolong reliable predictions. Comparisons of the PDE-Net with the Frozen-PDE-Net and Freed-PDE-Net demonstrate the importance of using learnable and yet partially constrained filters, which is new to the literature.
326
+
327
+ # 4 NUMERICAL STUDIES: DIFFUSION EQUATIONS WITH NONLINEAR SOURCE
328
+
329
+ When modeling physical processes like particle transportation or energy transfer, in addition to convection and diffusion, we have to consider source/sink terms. In some problems, the source/sink plays an important role. For example, when convection-diffusion equations are used to describe the distribution and flow of pollutants in water or atmosphere, identifying the intensity of pollution source is equivalent to finding the source term, which is important for environmental pollution control problems.
330
+
331
+ ![](images/703bbaf000f92a01779d69e3b8502b45b0ae34b67628d6d327fb66a6987f950a.jpg)
332
+ Figure 9: First row: the true coefficients of the equation. From the left to right are coefficients of $u$ , $u _ { x }$ , $u _ { y }$ , $u _ { x x }$ , $u _ { x y }$ and $u _ { y y }$ . Second row: the learned coefficients by the PDE-Net assuming the order of the PDE is $\leq 4$ (same as the third row of Figure 6). Third row: the learned coefficients by the PDE-Net assuming the order of the PDE is $\leq 2$ . Last row: the errors between true and learned coefficients v.s. number of $\delta t$ -blocks $( 1 , 2 , \ldots , 1 3 )$ for PDE-Net assuming the PDE is of order $\leq 4$ (orange) and $\leq 2$ (green).
333
+
334
+ ![](images/a74f148a5ae2f03fdcf366d3eff5e1abce7c144150f0bd671e4c37e8a2fe28f6.jpg)
335
+ Figure 10: The images of all the variable coefficients learned from the Freed-PDE-Net.
336
+
337
+ # 4.1 SIMULATED DATA, TRAINING AND TESTING
338
+
339
+ We consider a 2-dimensional linear diffusion equation with a nonlinear source on $\Omega = [ 0 , 2 \pi ] \times$ $[ 0 , 2 \pi ]$ ,
340
+
341
+ $$
342
+ \left\{ \begin{array} { l l } { \frac { \partial u } { \partial t } } & { = c \Delta u + f _ { s } ( u ) } \\ { u | _ { t = 0 } } & { = u _ { 0 } ( x , y ) , } \end{array} \right. \quad \mathrm { w i t h ~ } ( t , x , y ) \in [ 0 , 0 . 2 ] \times \Omega ,
343
+ $$
344
+
345
+ where $c = 0 . 3$ and $f _ { s } ( u ) = 1 5 \sin ( u )$ . The computation domain $\Omega$ is discretized using a $5 0 \times 5 0$ regular mesh. Data is generated by solving problem (11) using forward Euler for temporal discretization (with time step size $\delta t = 0 . 0 0 0 9 )$ and central differencing for spatial discretization on $1 0 0 \times 1 0 0$ mesh, and then restricted to the $5 0 \times 5 0$ mesh. We assume zero boundary condition and the initial value $u _ { 0 } ( x , y )$ is generated by $\begin{array} { r } { u _ { 0 } ( x , y ) = u _ { 0 } ^ { \prime } ( x , y ) \frac { x ( 2 \pi - x ) y ( 2 \pi - y ) } { ( 2 \pi ) ^ { 4 } } } \end{array}$ ) x(2π−x)y(2π−y)4 , where u0 is obtained from (9) with maximum allowable frequency $N = 6$ . Same as the numerical setting in Section 3, Gaussian noise is added to each sample sequence $u ( x , y , t ) , t \in [ 0 , 0 . 2 ]$ as described by (10).
346
+
347
+ Suppose we know a priori that the underlying PDE is a convection-diffusion equation of order no more than 2 with a nonlinear source depending on the variable $u$ . Then, the response function $F$ takes the following form
348
+
349
+ $$
350
+ F = \sum _ { 1 \leq i + j \leq 2 } f _ { i j } ( x , y ) \frac { \partial ^ { i + j } u } { \partial x ^ { i } \partial y ^ { j } } + f _ { s } ( u ) .
351
+ $$
352
+
353
+ Each $\delta t$ -block of the PDE-Net can be written as
354
+
355
+ $\check { \iota } ( t _ { n + 1 } , \cdot ) = D _ { 0 } u ( t _ { n } , \cdot ) + \delta t \cdot \left( c _ { 0 1 } D _ { 0 1 } u + c _ { 1 0 } D _ { 1 0 } u + c _ { 1 1 } D _ { 1 1 } u + c _ { 2 0 } D _ { 2 0 } u + c _ { 0 2 } D _ { 0 2 } u \right) + \tilde { f } _ { s } ( u ) ,$ where $\{ D _ { 0 } , D _ { i j } : 1 \le i + j \le 2 \}$ are convolution operators and $\{ c _ { i j } : 1 \le i + j \le 2 \}$ are 2D arrays which approximate functions $f _ { i j } ( x , y )$ on $\Omega$ . The approximation is achieved using piecewise quadratic polynomial interpolation with smooth transitions at the boundaries of each piece. The approximation of $\tilde { f } _ { s }$ is obtained by piecewise 4th order polynomial approximation over a regular grid of the interval $[ - 3 0 , 3 0 ]$ with 40 grid points. The training and testing strategy is exactly the same as in Section 3. In our experiments, the size of the filters is $7 \times 7$ . The total number of trainable parameters for each $\delta t$ -block is approximately $1 . 2 \mathrm { k }$ .
356
+
357
+ # 4.2 RESULTS AND DISCUSSIONS
358
+
359
+ This section presents numerical results of the trained PDE-Net using the data set described in Section 4.1. We will observe how the trained PDE-Net performs in terms of prediction of dynamical behavior and identification of the underlying PDE model.
360
+
361
+ # PREDICTING LONG-TIME DYNAMICS
362
+
363
+ We demonstrate the ability of the trained PDE-Net in prediction, which in the language of machine learning is the ability to generalize. The testing method is exactly the same as the method described in Section 3. Comparisons between PDE-Net and Frozen-PDE-Net are shown in Figure 11, where we can clearly see the advantage of learning the filters. Long-time predictions of the PDE-Net is shown in Figure 12 and we visualize the predicted dynamics in Figure 13. To further demonstrate how well the learned PDE-Net generalizes, we generate initial values following (9) with highest frequency equal to 10, followed by adding noise (10). Note that the maximum allowable frequency in the training set is only 6. The results of long-time prediction and the estimated dynamics are shown in Figure 14. All these results show that the learned PDE-Net performs well in prediction.
364
+
365
+ ![](images/4cc69e70f7e27514f586671b5902845dbdc69a2e30ee26ec51ac99eff744860b.jpg)
366
+ Figure 11: Prediction errors of the PDE-Net (orange) and Frozen-PDE-Net (blue) with $7 \times 7$ filters. In each plot, the horizontal axis indicates the time of prediction in the interval $( 0 , 0 . 6 ]$ , and the vertical axis shows the normalized errors. The banded curves indicate the $2 5 \%$ & $7 5 \%$ percentile of the normalized errors among 560 test samples.
367
+
368
+ # DISCOVERING THE HIDDEN EQUATION
369
+
370
+ For the PDE (11), identifying the PDE amounts to finding the coefficients $\{ c _ { i j } : 1 \le i + j \le 2 \}$ that approximate $\{ f _ { i j } : 1 \le i + j \le 2 \}$ , and $\tilde { f } _ { s }$ that approximates $f _ { s }$ . The computed coefficients $\{ c _ { i j } : 1 \le i + j \le 2 \}$ of the trained PDE-Net are shown in Figure 15, and the computed $\tilde { f } _ { s }$ is shown in Figure 16 (left). Note that the first order terms are absent from the PDE (11), and the corresponding coefficients learned by the PDE-Net are indeed close to zero. The approximation of $f _ { s }$ is more accurate near the center of the interval than near the boundary. This is because the value of $u$ in the data set is mostly distributed near the center (Figure 16(right)).
371
+
372
+ ![](images/dc6236172d9731eb393a57d88ac9a0d626787aea247a178916b2356485e0cdb3.jpg)
373
+ Figure 12: Long-time prediction for the PDE-Net with $7 \times 7$ filters in $( 0 , 2 ]$ .
374
+
375
+ ![](images/23aa70215427b22f83980687bc6fdbeb20d2c1a778237e7797801fd07601249d.jpg)
376
+ Figure 13: Images of the true dynamics and the predicted dynamics. The first row shows the images of the true dynamics. The second row shows the images of the predicted dynamics using the PDENet having $3 \delta t$ -blocks with $7 \times 7$ filters. Here, $\delta t = 0 . 0 1$ .
377
+
378
+ # 5 CONCLUSION AND DISCUSSION
379
+
380
+ In this paper, we designed a deep feed-forward network, called the PDE-Net, to discover the hidden PDE model from the observed dynamics and to predict the dynamical behavior. The PDE-Net consists of two major components which are jointly trained: to approximate differential operations by convolutions with properly constrained filters, and to approximate the nonlinear response by deep neural networks or other machine learning methods. The PDE-Net is suitable for learning PDEs as general as in (1). However, if we have a prior knowledge on the form of the response function $F$ , we can easily adjust the network architecture by taking advantage of the additional information. This may simplify the training and improve the results. As an example, we considered a linear variable-coefficient convection-diffusion equation. The results show that the PDE-Net can uncover the hidden equation of the observed dynamics, and predict the dynamical behavior for a relatively long time, even in a noisy environment. Furthermore, having deep structure (i.e. multiple $\delta t$ -blocks) and larger learnable filters can improve the PDE-Net in terms of stability and can prolong reliable predictions. As part of the future work, we will try the proposed framework on real data sets. One of the important directions is to uncover hidden variables which cannot be measured by sensors directly, such as in data assimilation. Another interesting direction which is worth exploring is to learn stable and consistent numerical schemes for a given PDE model based on the architecture of the PDE-Net.
381
+
382
+ # REFERENCES
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+ ![](images/c4653d2a2563e1fb92faae897d39d9b5d24b7525bf71858b37829874476368a3.jpg)
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+ Figure 14: Testing with higher frequency initializations (diffusion equation with a nonlinear source). First row: long-time prediction. Second row: estimated dynamics.Here, $\delta t = 0 . 0 1$ .
390
+
391
+ ![](images/88af0fe7caa396e77579707f8eac31836a383d720eaccee2af387afdd07ee58e.jpg)
392
+ Figure 15: First row: the true coefficients $\{ f _ { i j } : 1 \le i + j \le 2 \}$ of the equation. Second row: the learned coefficients $\{ c _ { i j } : 1 \le i + j \le 2 \}$ by the PDE-Net with 3 $\delta t$ -blocks and $7 \times 7$ filters.
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+ Figure 16: Left: the true source function $f _ { s }$ and estimated source function $\tilde { f } _ { s }$ . Right: distribution of the values of $u$ during training.
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+ Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Identity mappings in deep residual networks. In European Conference on Computer Vision, pp. 630–645. Springer, 2016b.
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