Taskonomy of Transfer Learning

Taskonomy of Transfer Learning

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  Taskonomy: Disentangling TaskTransfer Learning Amir R. Zamir1,2 Alexander Sax1∗ William Shen1∗ Leonidas Guibas1 Jitendra Malik2 Silvio Savarese1 1 Stanford University 2 University of California, Berkeley http://taskonomy.vision/ Abstract Do visual tasks have a relationship, or are they unre- lated? For instance, could having surface normals sim- plify estimating the depth of an image? Intuition answers these questions positively, implying existence of a structure among visual tasks. Knowing this structure has notable val- ues; it is the concept underlying transfer learning and pro- vides a principled way for identifying redundancies across tasks, in order to, for instance, seamlessly reuse supervi- sion among related tasks or solve many tasks in one system without piling up the complexity. We propose a fully computational approach for model- ing the structure of the space of visual tasks. This is done via finding (first and higher-order) transfer learning depen- dencies across a dictionary of twenty-six 2D, 2.5D, 3D, and semantic tasks in a latent space. The product is a computa- tional taxonomic map for task transfer learning. We study the consequences of this structure, e.g. nontrivial emerged relationships, and exploit them to reduce the demand for la- beled data. For example, we show that the total number of labeled datapoints needed for solving a set of 10 tasks can be reduced by roughly 2 3 (compared to training indepen- dently) while keeping the performance nearly the same. We provide a set of tools for computing and probing this taxo- nomical structure including a solver that users can employ to devise efficient supervision policies for their use cases. 1. Introduction Object recognition, depth estimation, edge detection, pose estimation, etc are examples of common vision tasks deemed useful and tackled by the research community. Some of them have rather clear relationships: we under- stand that surface normals and depth are related (one is a derivate of the other), or vanishing points in a room are use- ful for orientation and layout estimation. Other relation- ships are less clear: how edge detection and the shading in a room can, together, perform pose estimation. ∗Equal. Curvature Semantic Segm. Denoising Autoencoding Object Class. (1000 class) Z-Depth Distance 2D Keypoints 2D Segm. 2.5D Segm. Triplet Cam. Pose Cam. Pose (fixated) Cam. Pose (non-fixated) Vanishing Pts Room Layout 3D Edges Normals Point Matching Reshading Normals Point Matching 3D Edges Reshading Figure 1: A sample task structure discovered by the computational task taxonomy (taskonomy). It found that, for instance, by combining the learned features of a surface normal estimator and occlusion edge detector, good networks for reshading and point matching can be rapidly trained with little labeled data. The field of computer vision has indeed gone far without explicitly using these relationships. We have made remark- able progress by developing advanced learning machinery (e.g. ConvNets) capable of finding complex mappings from X to Y when many pairs of (x, y) s.t. x ∈ X, y ∈ Y are given as training data. This is usually referred to as fully su- pervised learning and often leads to problems being solved in isolation. Siloing tasks makes training a new task or a comprehensive perception system a Sisyphean challenge, whereby each task needs to be learned individually from scratch. Doing so ignores their quantifiably useful relation- ships leading to a massive labeled data requirement. Alternatively, a model aware of the relationships among tasks demands less supervision, uses less computation, and behaves in more predictable ways. Incorporating such a structure is the first stepping stone towards develop- 1
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  ing provably efficientcomprehensive/universal perception models [34, 4], i.e. ones that can solve a large set of tasks before becoming intractable in supervision or computation demands. However, this task space structure and its effects are still largely unknown. The relationships are non-trivial, and finding them is complicated by the fact that we have imperfect learning models and optimizers. In this paper, we attempt to shed light on this underlying structure and present a framework for mapping the space of visual tasks. Here what we mean by “structure” is a collection of com- putationally found relations specifying which tasks supply useful information to another, and by how much (see Fig. 1). We employ a fully computational approach for this pur- pose, with neural networks as the adopted computational function class. In a feedforward network, each layer succes- sively forms more abstract representations of the input con- taining the information needed for mapping the input to the output. These representations, however, can transmit statis- tics useful for solving other outputs (tasks), presumably if the tasks are related in some form [83, 19, 58, 46]. This is the basis of our approach: we compute an affinity matrix among tasks based on whether the solution for one task can be sufficiently easily read out of the representation trained for another task. Such transfers are exhaustively sampled, and a Binary Integer Programming formulation extracts a globally efficient transfer policy from them. We show this model leads to solving tasks with far less data than learn- ing them independently and the resulting structure holds on common datasets (ImageNet [78] and Places [104]). Being fully computational and representation-based, the proposed approach avoids imposing prior (possibly incor- rect) assumptions on the task space. This is crucial because the priors about task relations are often derived from either human intuition or analytical knowledge, while neural net- works need not operate on the same principles [63, 33, 40, 45, 102, 88]. For instance, although we might expect depth to transfer to surface normals better (derivatives are easy), the opposite is found to be the better direction in a compu- tational framework (i.e. suited neural networks better). An interactive taxonomy solver which uses our model to suggest data-efficient curricula, a live demo, dataset, and code are available at http://taskonomy.vision/. 2. Related Work Assertions of existence of a structure among tasks date back to the early years of modern computer science, e.g. with Turing arguing for using learning elements [95, 98] rather than the final outcome or Jean Piaget’s works on developmental stages using previously learned stages as sources [74, 39, 38], and have extended to recent works [76, 73, 50, 18, 97, 61, 11, 66]. Here we make an attempt to actu- ally find this structure. We acknowledge that this is related to a breadth of topics, e.g. compositional modeling [35, 10, 13, 23, 55, 92, 90], homomorphic cryptography [42], life- long learning [93, 15, 85, 84], functional maps [71], certain aspects of Bayesian inference and Dirichlet processes [54, 91, 90, 89, 37, 39], few-shot learning [81, 25, 24, 70, 86], transfer learning [75, 84, 29, 64, 67, 59], un/semi/self- supervised learning [22, 8, 17, 103, 19, 83], which are stud- ied across various fields [73, 94, 12]. We review the topics most pertinent to vision within the constraints of space: Self-supervised learning methods leverage the inherent relationships between tasks to learn a desired expensive one (e.g. object detection) via a cheap surrogate (e.g. coloriza- tion) [68, 72, 17, 103, 100, 69]. Specifically, they use a manually-entered local part of the structure in the task space (as the surrogate task is manually defined). In contrast, our approach models this large space of tasks in a computational manner and can discover obscure relationships. Unsupervised learning is concerned with the redundan- cies in the input domain and leveraging them for forming compact representations, which are usually agnostic to the downstream task [8, 49, 20, 9, 32, 77]. Our approach is not unsupervised by definition as it is not agnostic to the tasks. Instead, it models the space tasks belong to and in a way utilizes the functional redundancies among tasks. Meta-learning generally seeks performing the learning at a level higher than where conventional learning occurs, e.g. as employed in reinforcement learning [21, 31, 28], optimization [2, 82, 48], or certain architectural mecha- nisms [27, 30, 87, 65]. The motivation behind meta learn- ing has similarities to ours and our outcome can be seen as a computational meta-structure of the space of tasks. Multi-task learning targets developing systems that can provide multiple outputs for an input in one run [50, 18]. Multi-task learning has experienced recent progress and the reported advantages are another support for existence of a useful structure among tasks [93, 100, 50, 76, 73, 50, 18, 97, 61, 11, 66]. Unlike multi-task learning, we explicitly model the relations among tasks and extract a meta-structure. The large number of tasks we consider also makes developing one multi-task network for all infeasible. Domain adaption seeks to render a function that is de- veloped on a certain domain applicable to another [44, 99, 5, 80, 52, 26, 36]. It often addresses a shift in the input do- main, e.g. webcam images to D-SLR [47], while the task is kept the same. In contrast, our framework is concerned with output (task) space, hence can be viewed as task/output adaptation. We also perform the adaptation in a larger space among many elements, rather than two or a few. In the context of our approach to modeling transfer learn- ing across tasks: Learning Theoretic approaches may overlap with any of the above topics and usually focus on providing gener- alization guarantees. They vary in their approach: e.g. by modeling transferability with the transfer family required 2
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  2nd 3rd Frozen 1st Order Order Order Task-specific 2D Segm. 3DKeypoints 2.5D Segm Normals ReshadingLayout 2D Segm. 3D Keypoints 2.5D Segm Normals ReshadingLayout (I) Task-specific Modeling (II) Transfer Modeling (III) Task Affinity Normalization (IV) Compute Taxonomy Output space Task Space (representation) Input space Object Class. (100) Object Class. (1000) Curvature Scene Class. 0) Semantic Segm. Normals 3D Keypoints Denoising Autoencoding 2D Segm. 2D Edges 2D Keypoints In-painting Colorization Matching 2.5D Segm. Z-Depth Distance ass. Egomotion Cam. Pose (fix) Occlusion Edges Reshading Cam. Pose (nonfix) Layout Cam. Pose Vanishing Pts. JigsawJigsawJigsawJigsawJigsawJigsawJigsawJigsaw Randomm Proj. Binary Integer Program AHP task affinities . . . . . . Figure 2: Computational modeling of task relations and creating the taxonomy. From left to right: I. Train task-specific networks. II. Train (first order and higher) transfer functions among tasks in a latent space. III. Get normalized transfer affinities using AHP (Analytic Hierarchy Process). IV. Find global transfer taxonomy using BIP (Binary Integer Program). to map a hypothesis for one task onto a hypothesis for an- other [7], through information-based approaches [60], or through modeling inductive bias [6]. For these guarantees, learning theoretic approaches usually rely on intractable computations, or avoid such computations by restricting the model or task. Our method draws inspiration from theoreti- cal approaches but eschews (for now) theoretical guarantees in order to use modern neural machinery. 3. Method We define the problem as follows: we want to max- imize the collective performance on a set of tasks T = {t1, ..., tn}, subject to the constraint that we have a limited supervision budget γ (due to financial, computational, or time constraints). We define our supervision budget γ to be the maximum allowable number of tasks that we are willing to train from scratch (i.e. source tasks). The task dictionary is defined as V=T ∪ S where T is the set of tasks which we want solved (target), and S is the set of tasks that can be trained (source). Therefore, T − T ∩ S are the tasks that we want solved but cannot train (“target-only”), T ∩ S are the tasks that we want solved but could play as source too, and S − T ∩ S are the “source-only” tasks which we may not directly care about to solve (e.g. jigsaw puzzle) but can be optionally used if they increase the performance on T . The task taxonomy (taskonomy) is a computationally found directed hypergraph that captures the notion of task transferability over any given task dictionary. An edge be- tween a group of source tasks and a target task represents a feasible transfer case and its weight is the prediction of its performance. We use these edges to estimate the globally optimal transfer policy to solve T . Taxonomy produces a family of such graphs, parameterized by the available su- pervision budget, chosen tasks, transfer orders, and transfer functions’ expressiveness. Taxonomy is built using a four step process depicted in Fig. 2. In stage I, a task-specific network for each task in S Query Image AutoencodingIn-painting Object Class. Scene Class. Jigsaw puzzle Colorization 2D Segm. 2.5D Segm. Semantic Segm. Vanishing Points 2D Edges 3D Edges 2D Keypoints 3D Keypoints 3D Curvature Image Reshading Denoising Cam. Pose (non-fixated) Cam. Pose(fixated) Triplet Cam. Pose Room Layout Point Matching Top 5 prediction: sliding door home theater, home theatre studio couch, day bed china cabinet, china closet entertainment center Eucl. DistanceSurface Normals Top 2 prediction: living room television room Figure 3: Task Dictionary. Outputs of 24 (of 26) task-specific networks for a query (top left). See results of applying frame-wise on a video here. is trained. In stage II, all feasible transfers between sources and targets are trained. We include higher-order transfers which use multiple inputs task to transfer to one target. In stage III, the task affinities acquired from transfer function performances are normalized, and in stage IV, we synthe- size a hypergraph which can predict the performance of any transfer policy and optimize for the optimal one. A vision task is an abstraction read from a raw image. We denote a task t more formally as a function ft which maps image I to ft(I). Our dataset, D, contains for each task t a set of training pairs (I, ft(I)), e.g. (image, depth). 3
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  Source Task EncoderTarget Task Output (e.g., curvature)Frozen Representation Transfer Function 2nd order 3rd order ... (e.g., surface normal) I sE s→tD sE (I) Figure 4: Transfer Function. A small readout function is trained to map representations of source task’s frozen encoder to target task’s labels. If order> 1, transfer function receives representations from multiple sources. Task Dictionary: Our mapping of task space is done via (26) tasks included in the dictionary, so we ensure they cover common themes in computer vision (2D, 3D, seman- tics, etc) to elucidate fine-grained structures of task space. See Fig. 3 for some of the tasks with detailed definition of each task provided in the supplementary material. We in- clude tasks with various levels of abstraction, ranging from solvable by a simple kernel convolved over the image (e.g. edge detection) to tasks requiring basic understanding of scene geometry (e.g. vanishing points) and more abstract ones involving semantics (e.g. scene classification). It is critical to note the task dictionary is meant to be a sampled set, not an exhaustive list, from a denser space of all conceivable visual tasks/abstractions. Sampling gives us a tractable way to sparsely model a dense space, and the hy- pothesis is that (subject to a proper sampling) the derived model should generalize to out-of-dictionary tasks. The more regular / better sampled the space, the better the gen- eralization. We evaluate this in Sec. 4.2 with supportive results. For evaluation of the robustness of results w.r.t the choice of dictionary, see the supplementary material. Dataset: We need a dataset that has annotations for ev- ery task on every image. Training all of our tasks on exactly the same pixels eliminates the possibility that the observed transferabilities are affected by different input data pecu- liarities rather than only task intrinsics. There has not been such a dataset of scale made of real images, so we created a dataset of 4 million images of indoor scenes from about 600 buildings; every image has an annotation for every task. The images are registered on and aligned with building- wide meshes similar to [3, 101, 14] enabling us to program- matically compute the ground truth for many tasks without human labeling. For the tasks that still require labels (e.g. scene classes), we generate them using Knowledge Distil- lation [43] from known methods [104, 57, 56, 78]. See the supplementary material for full details of the process and a user study on the final quality of labels generated using Knowledge Distillation (showing < 7% error). 3.1. Step I: Task-Specific Modeling We train a fully supervised task-specific network for each task in S. Task-specific networks have an encoder- decoder architecture homogeneous across all tasks, where Layout Layout ScratchReshade Input Ground Truth Task Specific TransfersResults(2ktrainingimages) ScratchReshade Input Ground Truth Task Specific 2.5D Segmentation SurfaceNormal Estimation Autoenc. Autoenc. 2D Segm. 2D Segm. Figure 5: Transfer results to normals (upper) and 2.5D Segmentation (lower) from 5 different source tasks. The spread in transferability among different sources is apparent, with reshading among top-performing ones in this case. Task-specific networks were trained on 60x more data. “Scratch” was trained from scratch without transfer learning. the encoder is large enough to extract powerful represen- tations, and the decoder is large enough to achieve a good performance but is much smaller than the encoder. 3.2. Step II: Transfer Modeling Given a source task s and a target task t, where s ∈ S and t ∈ T , a transfer network learns a small readout func- tion for t given a statistic computed for s (see Fig 4). The statistic is the representation for image I from the encoder of s: Es(I). The readout function (Ds→t) is parameterized by θs→t minimizing the loss Lt: Ds→t := arg min θ EI∈D Lt Dθ Es(I) , ft(I) , (1) where ft(I) is ground truth of t for image I. Es(I) may or may not be sufficient for solving t depending on the relation between t and s (examples in Fig. 5). Thus, the performance of Ds→t is a useful metric as task affinity. We train transfer functions for all feasible source-target combinations. Accessibility: For a transfer to be successful, the latent representation of the source should both be inclusive of suf- ficient information for solving the target and have the in- formation accessible, i.e. easily extractable (otherwise, the raw image or its compression based representations would be optimal). Thus, it is crucial for us to adopt a low-capacity (small) architecture as transfer function trained with a small amount of data, in order to measure transferability condi- tioned on being highly accessible. We use a shallow fully convolutional network and train it with little data (8x to 120x less than task-specific networks). Higher-Order Transfers: Multiple source tasks can contain complementary information for solving a target task (see examples in Fig 6). We include higher-order transfers which are the same as first order but receive multiple rep- resentations in the input. Thus, our transfers are functions D : ℘(S) → T , where ℘ is the powerset operator. As there is a combinatorial explosion in the number of feasible higher-order transfers (|T | × |S| k for kth order), we employ a sampling procedure with the goal of filtering out higher-order transfers that are less likely to yield good results, without training them. We use a beam search: for 4
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  Image GT(Normals) FullySupervised Image GT(Reshade) Fully Supervised { 3D Keypoints Surface Normals } 2 nd order transfer+ ={Occlusion Edges Curvature } 2 nd order transfer+ = Figure 6: Higher-Order Transfers. Representations can contain com- plementary information. E.g. by transferring simultaneously from 3D Edges and Curvature individual stairs were brought out. See our publicly available interactive transfer visualization page for more examples. transfers of order k ≤ 5 to a target, we select its 5 best sources (according to 1st order performances) and include all of their order-k combination. For k ≥ 5, we use a beam of size 1 and compute the transfer from the top k sources. Transitive Transfers: We examined if transitive task transfers (s → t1 → t2) could improve the performance over their direct counterpart (s → t2), but found that the two had equal performance in almost all cases in both high- data and low-data scenarios. The experiment is provided in the supplementary material. Therefore, we need not con- sider the cases where branching would be more than one level deep when searching for the optimal transfer path. 3.3. Step III: Ordinal Normalization using Analytic Hierarchy Process (AHP) We want to have an affinity matrix of transferabilities across tasks. Aggregating the raw losses/evaluations Ls→t from transfer functions into a matrix is obviously problem- atic as they have vastly different scales and live in different spaces (see Fig. 7-left). Hence, a proper normalization is needed. A naive solution would be to linearly rescale each row of the matrix to the range [0, 1]. This approach fails when the actual output quality increases at different speeds w.r.t. the loss. As the loss-quality curve is generally un- known, such approaches to normalization are ineffective. Instead, we use an ordinal approach in which the output quality and loss are only assumed to change monotonically. For each t, we construct Wt a pairwise tournament matrix between all feasible sources for transferring to t. The ele- ment at (i, j) is the percentage of images in a held-out test set, Dtest, on which si transfered to t better than sj did (i.e. Dsi→t(I) > Dsj →t(I)). We clip this intermediate pairwise matrix Wt to be in [0.001, 0.999] as a form of Laplace smoothing. Then we divide Wt = Wt/WT t so that the matrix shows how many times better si is compared to sj. The final tournament ratio matrix is positive reciprocal with each element wi,j of Wt : wi,j = EI∈Dtest [Dsi→t(I) > Dsj →t(I)] EI∈Dtest [Dsi→t(I) < Dsj →t(I)] . (2) We quantify the final transferability of si to t as the cor- Autoencoding SceneClass Curvature Denoising 2DEdges OcclusionEdges 2DKeypoint 3DKeypoint ReshadingZ-DepthDistanceNormals Egomotion VanishingPts. 2DSegm. 2.5DSegm. Cam.Pose(fix) Cam.Pose(nonfix)LayoutMatching SemanticSegm. Jigsaw In-PaintingColorization Random Proj. Task-Specific ObjectClass(100) AutoencodingSceneClass ObjectClass(100) Colorization Curvature Denoising OcclusionEdges 2DEdgesEgomotion Cam.Pose(fix) In-PaintingJigsaw 2DKeypoint 3DKeypoint Cam.Pose(nonfix) Matching Random Proj. ReshadingZ-DepthDistance 2DSegm. 2.5DSegm. Layout Normals SemanticSegm.Task-Specific VanishingPts. Autoencoding Object Class. (1000) Scene Class Curvature Occlusion Edges Egomotion Cam. Pose (fix) 2D Keypoint Layout Matching 2D Segm. Distance 2.5D Segm. Z-Depth Normals 3D Keypoint Denoising 2D Edges Cam. Pose (nonfix) Reshading Semantic Segm. Vanishing Pts. ObjectClass.(1000) ObjectClass.(1000) Figure 7: First-order task affinity matrix before (left) and after (right) Analytic Hierarchy Process (AHP) normalization. Lower means better transfered. For visualization, we use standard affinity-distance method dist = e−β·P (where β = 20 and e is element-wise matrix exponential). See supplementary material for the full matrix with higher-order transfers. responding (ith ) component of the principal eigenvector of Wt (normalized to sum to 1). The elements of the principal eigenvector are a measure of centrality, and are proportional to the amount of time that an infinite-length random walk on Wt will spend at any given source [62]. We stack the prin- cipal eigenvectors of Wt for all t ∈ T , to get an affinity matrix P (‘p’ for performance)—see Fig. 7, right. This approach is derived from Analytic Hierarchy Pro- cess [79], a method widely used in operations research to create a total order based on multiple pairwise comparisons. 3.4. Step IV: Computing the Global Taxonomy Given the normalized task affinity matrix, we need to devise a global transfer policy which maximizes collective performance across all tasks, while minimizing the used su- pervision. This problem can be formulated as subgraph se- lection where tasks are nodes and transfers are edges. The optimal subgraph picks the ideal source nodes and the best edges from these sources to targets while satisfying that the number of source nodes does not exceed the supervi- sion budget. We solve this subgraph selection problem us- ing Boolean Integer Programming (BIP), described below, which can be solved optimally and efficiently [41, 16]. Our transfers (edges), E, are indexed by i with the form ({si 1, . . . , si mi }, ti ) where {si 1, . . . , si mi } ⊂ S and ti ∈ T . We define operators returning target and sources of an edge: {si 1, . . . , si mi }, ti sources −−−−−→ {si 1, . . . , si mi } {si 1, . . . , si mi }, ti target −−−−→ ti . Solving a task t by fully supervising it is denoted as {t}, t . We also index the targets T with j so that in this section, i represents edges and j represents targets. The parameters of the problem are: the supervision bud- get (γ) and a measure of performance on a target from each of its transfers (pi), i.e. the affinities from P. We can also optionally include additional parameters of: rj specifying the relative importance of each target task and i specifying the relative cost of acquiring labels for each task. 5
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  The BIP isparameterized by a vector x where each trans- fer and each task is represented by a binary variable; x indi- cates which nodes are picked to be source and which trans- fers are selected. The canonical form for a BIP is: maximize: cT x , subject to: Ax b and x ∈ {0, 1}|E|+|V| . Each element ci for a transfer is the product of the im- portance of its target task and its transfer performance: ci := rtarget(i) · pi . (3) Hence, the collective performance on all targets is the sum- mation of their individual AHP performance, pi, weighted by the user specified importance, ri. Now we add three types of constraints via matrix A to enforce each feasible solution of the BIP instance corre- sponds to a valid subgraph for our transfer learning prob- lem: Constraint I: if a transfer is included in the subgraph, all of its source nodes/tasks must be included too, Con- straint II: each target task has exactly one transfer in, Con- straint III: supervision budget is not exceeded. Constraint I: For each row ai in A we require ai ·x ≤ bi, where ai,k :=    |sources(i)| if k = i −1 if (k − |E|) ∈ sources(i) 0 otherwise (4) bi = 0. (5) Constraint II: Via two rows a|E|+j and a|E|+j+1, we en- force that target j has exactly one transfer: a|E|+j,i := 1{target(i)=j}, b|E|+j := +1, a|E|+j+1,i := −1{target(i)=j}, b|E|+j+1 := −1. (6) Constraint III: the solution is enforced to not exceed the budget. Each transfer i is assigned a label cost i, so a|E|+2|V|+1,i := i, b|E|+2|V|+1 := γ. (7) The elements of A not defined above are set to 0. The problem is now a valid BIP and can be optimally solved in a fraction of a second [41]. The BIP solution ˆx corresponds to the optimal subgraph, which is our taxonomy. 4. Experiments With 26 tasks in the dictionary (4 source-only tasks), our approach leads to training 26 fully supervised task-specific networks, 22 × 25 transfer networks in 1st order, and 22 × 25 k for kth order, from which we sample according to the procedure in Sec. 3. The total number of transfer functions trained for the taxonomy was ∼3,000 which took 47,886 GPU hours on the cloud. Out of 26 tasks, we use the following 4 as source-only tasks (described in Sec. 3) in the experiments: colorization, jigsaw puzzle, in-painting, random projection. However, the method is applicable to an arbitrary partitioning of the dictionary into T and S. The interactive solver website al- Task avg rand Task avg rand Task avg rand Denoising 100 99.9 Layout 99.6 89.1 Scene Class. 97.0 93.4 Autoenc. 100 99.8 2D Edges 100 99.9 Occ. Edges 100 95.4 Reshading 94.9 95.2 Pose (fix) 76.3 79.5 Pose (nonfix) 60.2 61.9 Inpainting 99.9 - 2D Segm. 97.7 95.7 2.5D Segm. 94.2 89.4 Curvature 78.7 93.4 Matching 86.8 84.6 Egomotion 67.5 72.3 Normals 99.4 99.5 Vanishing 99.5 96.4 2D Keypnt. 99.8 99.4 Z-Depth 92.3 91.1 Distance 92.4 92.1 3D Keypnt. 96.0 96.9 Mean 92.4 90.9 Table 1: Task-Specific Networks’ Sanity: Win rates vs. random (Gaus- sian) network representation readout and statistically informed guess avg. lows the user to specify any desired partition. Network Architectures: We preserved the architectural and training details across tasks as homogeneously as possi- ble to avoid injecting any bias. The encoder architecture is identical across all task-specific networks and is a fully con- volutional ResNet-50 without pooling. All transfer func- tions include identical shallow networks with 2 conv layers (concatenated channel-wise if higher-order). The loss (Lt) and decoder’s architecture, though, have to depend on the task as the output structures of different tasks vary; for all pixel-to-pixel tasks, e.g. normal estimation, the decoder is a 15-layer fully convolutional network; for low dimensional tasks, e.g. vanishing points, it consists of 2-3 FC layers. All networks are trained using the same hyperparameters regardless of task and on exactly the same input images. Tasks with more than one input, e.g. relative camera pose, share weights between the encoder towers. Transfer net- works are all trained using the same hyperparameters as the task-specific networks, except that we anneal the learning rate earlier since they train much faster. Detailed definitions of architectures, training process, and experiments with dif- ferent encoders can be found in the supplementary material. Data Splits: Our dataset includes 4 million images. We made publicly available the models trained on full dataset, but for the experiments reported in the main paper, we used a subset of the dataset as the extracted structure stabi- lized and did not change when using more data (explained in Sec. 4.5). The used subset is partitioned into training (120k), validation (16k), and test (17k) images, each from non-overlapping sets of buildings. Our task-specific net- works are trained on the training set and the transfer net- works are trained on a subset of validation set, ranging from 1k images to 16k, in order to model the transfer patterns un- der different data regimes. In the main paper, we report all results under the 16k transfer supervision regime (∼10% of the split) and defer the additional sizes to the supplementary material and website (see Sec. 4.5). Transfer functions are evaluated on the test set. How good are the trained task-specific networks? Win rate (%) is the proportion of test set images for which a baseline is beaten. Table 1 provides win rates of the task- specifc networks vs. two baselines. Visual outputs for a ran- 6
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  Denoising Autoencoding 2D Edges 2D Keypoints 2DSegm. Normals Object Class. (1000) Scene Class. Curvature Occlusion Edges Egomotion Cam. Pose (fix) 3D Keypoints Cam. Pose (nonfix) Matching Reshading Z-Depth Distance Layout 2.5D Segm. Semantic Segm. Vanishing Pts. Coloooorizaaaation In-painttttingggg Jiggggsawwww Randoooom PPProjo . 3D Keypoints Autoencoding Object Class. (1000) Denoising 2D Keypoints Matching Normals Z-Depth Distance 2.5D Segm. Scene Class. 2D Edges Egomotion Cam. Pose (fix) Curvature Occlusion EdgesReshading Semantic Segm. Cam. Pose (nonfix) Layout 2D Segm. Vanishing Pts. Colorrriiiizattttiiiion In-ppppainnnnting Jiiiiggggsawwww Randdddom Projo . Scene Class. Object Class. (1000) Denoising Autoencoding 2D Keypoints 2D Segm. 2D Edges Normals Curvature Occlusion Edges Egomotion Cam. Pose (fix) 3D Keypoints Cam. Pose (nonfix) Matching Reshading Z-Depth Distance Layout 2.5D Segm. Semantic Segm. Vanishing Pts. Coloooorizaaaation Jiggggsawwww Randoooom Projo . Scene Class. Object Class. (1000) Denoising Autoencoding 2D Segm. 2D Edges 2D Keypoints In-painting 3D Keypoints Matching Normals Z-Depth Distance 2.5D Segm. Egomotion Cam. Pose (fix) Curvature Occlusion Edges Reshading Semantic Segm. Cam. Pose (nonfix) Layout Vanishing Pts. Coloooorrrizaaaation Jigsaaaaw Randoooommmm Projo . Object Class. (1000) Curvature Scene Class. Semantic Segm. Normals 3D Keypoints Denoising Autoencoding 2D Segm. 2D Edges 2D Keypoints In-painting Colorization Matching 2.5D Segm. Z-Depth Distance Egomotion Cam. Pose (fix) Occlusion Edges Reshading Cam. Pose (nonfix) Layout Vanishing Pts. iiiggggsaaaaw RRRRaaaaannnndom Projo . 2D Segm. Object Class. (1000) Matching 2.5D Segm. Z-Depth Distance Normals Occlusion Edges Egomotion Cam. Pose (fix) Cam. Pose (nonfix) Reshading Layout Semantic Segm. Collllorization Denoising 2D Edges In-ppppaintttting JJJJig ii ssssaw 2D Keypoints RRRaaaannnndom Projo . Vanishing Pts. Curvature Autoencoding Scene Class. 3D Keypoints . Object Class. (1000) Curvature 2D Keypoints 2D Segm. Denoising 3D Keypoints 2.5D Segm. Matching Normals Distance Z-Depth Occlusion Edges Semantic Segm. Cam. Pose (nonfix) Egomotion Cam. Pose (fix) Reshading Vanishing Pts. Layout Autoencoding Scene Class. Coloooorizaaaation 2D Edges In-painting Jiiiiggggsaaaawwww Randommm PPPProjo ....... Autoencoding 2D Segm. Denoising 2D Edges Object Class. (1000) Curvature Semantic Segm. Normals 3D Keypoints Matching 2.5D Segm. Distance Z-Depth Occlusion Edges Cam. Pose (nonfix) Egomotion Cam. Pose (fix) Reshading Layout Vanishing Pts. Scene Class. Coloooorrizaaaation In-ppppainnnntttting Jiggggsaaaaw 2D Keypoints andoooommmm Projo . 2.5D Segm. Egomotion Autoencoding Object Class. (1000) Scene Class. Colorizattttion Curvature Denoising 2D Edges Occlusion Edges Cam. Pose (fix) In-ppppainttttinggg Jigggsaw 2D Keypoints Cam. Pose (nonfix) Matching Randdddom Projo . Reshading Z-Depth Distance Layout 2D Segm. Semantic Segm. Vanishing Pts. 3D Keypoints Normals 3D Keypoints 2.5D Segm.Curvature Normals Egomotion Cam. Pose (fix) Semantic Segm. Autoencoding Object Class. (1000) Scene Class. Coloooorizaaaation Denoising 2D Edges Occlusion Edges In-ppppaintttting Jiiiigsawwww 2D Keypoints Cam. Pose (nonfix) Matching Randdddom Projo . Reshading Z-Depth Distance Layout 2D Segm. Vanishing Pts. Curvature Semantic Segm. Normals 3D Keypoints Reshading 2.5D Segm. Occlusion Edges Egomotion Cam. Pose (fix) Layout Cam. Pose (nonfix) Vanishing Pts. Autoencoding Object Class. (1000) Scene Class. CCCColoooorizattiiiooon Denoising 2D Edges In-painttttiiiing Jiiiigsawwww 2D Keypoints Matching RRannnndom Projo . Z-Depth Distance 2D Segm. Supervision Budget 2 TransferOrder1TransferOrder2TransferOrder4 Supervision Budget 8 Supervision Budget 15 Supervision Budget 26 Object Class. (1000) Curvature Scene Class. Semantic Segm. Normals 3D Keypoints Denoising Autoencoding 2D Segm. 2D Edges 2D Keypoints In-painting Colorization Matching 2.5D Segm. Z-Depth Distance Egomotion Cam. Pose (fix) Occlusion Edges Reshading Cam. Pose (nonfix) Layout Vanishing Pts. Jigssssaw RRRRaannnndom Projo . Supervision Budget 8 - Order 4 (zoomed) 3D Keypoints Autoencoding Object Class. (1000) Denoising 2D Keypoints Matching Normals Z-Depth Distance 2.5D Segm. Scene Class 2D Edges Egomotion Cam. Pose (fix) Curvature Occlusion EdgesReshading Semantic Segm. Cam. Pose (nonfix) Layout 2D Segm. Vanishing Pts. Colorriiiizattttiiiion In-ppppainnnnting Randdddom Projo . Jiiiiggggsawwww In-painttttingggg Figure 8: Computed taxonomies for solving 22 tasks given various supervision budgets (x-axes), and maximum allowed transfer orders (y-axes). One is magnified for better visibility. Nodes with incoming edges are target tasks, and the number of their incoming edges is the order of their chosen transfer function. Still transferring to some targets when tge budget is 26 (full budget) means certain transfers started performing better than their fully supervised task-specific counterpart. See the interactive solver website for color coding of the nodes by Gain and Quality metrics. Dimmed nodes are the source-only tasks, and thus, only participate in the taxonomy if found worthwhile by the BIP optimization to be one of the sources. dom test sample are in Fig. 3. The high win rates in Table 1 and qualitative results show the networks are well trained and stable and can be relied upon for modeling the task space. See results of applying the networks on a YouTube video frame-by-frame here. A live demo for user uploaded queries is available here. To get a sense of the quality of our networks vs. state-of- the-art task-specific methods, we compared our depth esti- mator vs. released models of [53] which led to outperform- ing [53] with a win rate of 88% and losses of 0.35 vs. 0.47 (further details in the supplementary material). In general, we found the task-specific networks to perform on par or better than state-of-the-art for many of the tasks, though we do not formally benchmark or claim this. 4.1. Evaluation of Computed Taxonomies Fig. 8 shows the computed taxonomies optimized to solve the full dictionary, i.e. all tasks are placed in T and S (except for 4 source-only tasks that are in S only). This was done for various supervision budgets (columns) and max- imum allowed order (rows) constraints. Still seeing trans- fers to some targets when the budget is 26 (full dictionary) means certain transfers became better than their fully super- vised task-specific counterpart. While Fig. 8 shows the structure and connectivity, Fig. 9 quantifies the results of taxonomy recommended transfer policies by two metrics of Gain and Quality, defined as: Gain Quality max transfer order=1 max transfer order=4 max transfer order=1 max transfer order=4 Supervision Budget Increase (→) Budget Figure 9: Evaluation of taxonomy computed for solving the full task dictionary. Gain (left) and Quality (right) values for each task using the policy suggested by the computed taxonomy, as the supervision budget increases(→). Shown for transfer orders 1 and 4. Gain: win rate (%) against a network trained from scratch using the same training data as transfer networks’. That is, the best that could be done if transfer learning was not utilized. This quantifies the gained value by transferring. Quality: win rate (%) against a fully supervised network trained with 120k images (gold standard). Red (0) and Blue (1) represent outperforming the ref- erence method on none and all of test set images, respec- tively (so the transition Red→White→Blue is desirable. White (.5) represents equal performance to reference). 7
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  Order Increase (→) OrderTask scratch ImageNet[51] Wang.[96] Agrawal.[1] Zamir.[100] Zhang.[103] Noroozi.[68] fullsup. Taxonomy Depth 88 88 93 89 88 84 86 43 - .03 .04 .04 .03 .04 .03 .03 .02 .02 Scene Cls. 80 52 83 74 74 71 75 15 - 3.30 2.76 3.56 3.15 3.17 3.09 3.19 2.23 2.63 Sem. Segm. 78 79 82 85 76 78 84 21 - 1.74 1.88 1.92 1.80 1.85 1.74 1.71 1.42 1.53 Object Cls. 79 54 82 76 75 76 76 34 - 4.08 3.57 4.27 3.99 3.98 4.00 3.97 3.26 3.46 Normals 97 98 98 98 98 97 97 6 - .22 .30 .34 .28 .28 .23 .24 .12 .15 2.5D Segm. 80 93 92 89 90 84 87 40 - .21 .34 .34 .26 .29 .22 .24 .16 .17 Occ. Edges 93 96 95 93 94 93 94 42 - .16 .19 .18 .17 .18 .16 .17 .12 .13 Curvature 88 94 89 85 88 92 88 29 - .25 .28 .26 .25 .26 .26 .25 .21 .22 Egomotion 79 78 83 77 76 74 71 59 - 8.60 8.58 9.26 8.41 8.34 8.15 7.94 7.32 6.85 Layout 80 76 85 79 77 78 70 36 - .66 .66 .85 .65 .65 .62 .54 .37 .41 Figure 10: Generalization to Novel Tasks. Each row shows a novel test task. Left: Gain and Quality values using the devised “all-for-one” transfer policies for novel tasks for orders 1-4. Right: Win rates (%) of the transfer policy over various self-supervised methods, ImageNet features, and scratch are shown in the colored rows. Note the large margin of win by taxonomy. The uncolored rows show corresponding loss values. Each column in Fig. 9 shows a supervision budget. As apparent, good results can be achieved even when the super- vision budget is notably smaller than the number of solved tasks, and as the budget increases, results improve (ex- pected). Results are shown for 2 maximum allowed orders. 4.2. Generalization to Novel Tasks The taxonomies in Sec. 4.1 were optimized for solving all tasks in the dictionary. In many situations, a practitioner is interested in a single task which even may not be in the dictionary. Here we evaluate how taxonomy transfers to a novel out-of-dictionary task with little data. This is done in an all-for-one scenario where we put one task in T and all others in S. The task in T is target-only and has no task-specific network. Its limited data (16k) is used to train small transfer networks to sources. This basi- cally localizes where the target would be in the taxonomy. Fig. 10 (left) shows the Gain and Quality of the transfer policy found by the BIP for each task. Fig. 10 (right) com- pares the taxonomy suggested policy against some of the best existing self-supervised methods [96, 103, 68, 100, 1], ImageNet FC7 features [51], training from scratch, and a fully supervised network (gold standard). The results in Fig. 10 (right) are noteworthy. The large win margin for taxonomy shows that carefully selecting transfer policies depending on the target is superior to fixed transfers, such as the ones employed by self-supervised methods. ImageNet features which are the most popular off-the-shelf features in vision are also outperformed by those policies. Additionally, though the taxonomy transfer policies lose to fully supervised networks (gold standard) in most cases, the results often get close with win rates in 40% range. These observations suggests the space has a rather Taxonomy Significance Test Supervision Budget Supervision Budget 5 7 9 3 1 Figure 11: Structure Significance. Our taxonomy compared with ran- dom transfer policies (random feasible taxonomies that use the maximum allowable supervision budget). Y-axis shows Quality or Gain, and X-axis is the supervision budget. Green and gray represent our taxonomy and ran- dom connectivities, respectively. Error bars denote 5th–95th percentiles. predicable and strong structure. For graph visualization of the all-for-one taxonomy policies please see the supplemen- tary material. The solver website allows generating the tax- onomy for arbitrary sets of target-only tasks. 4.3. Significance Test of the Structure The previous evaluations showed good transfer results in terms of Quality and Gain, but how crucial is it to use our taxonomy to choose smart transfers over just choosing any transfer? In other words, how significant/strong is the dis- covered structure of task space? Fig. 11 quantifies this by showing the performance of our taxonomy versus a large set of taxonomies with random connectivities. Our taxonomy outperformed all other connectivities by a large margin sig- nifying both existence of a strong structure in the space as well as a good modeling of it by our approach. Complete experimental details is available in supplementary material. 4.4. Evaluation on MIT Places & ImageNet To what extent are our findings dataset dependent, and would the taxonomy change if done on another dataset? We examined this by finding the ranking of all tasks for trans- ferring to two target tasks of object classification and scene classification on our dataset. We then fine tuned our task- specific networks on other datasets (MIT Places [104] for scene classification, ImageNet [78] for object classification) and evaluated them on their respective test sets and metrics. Fig. 12 shows how the results correlate with taxonomy’s ranking from our dataset. The Spearman’s rho between the taxonomy ranking and the Top-1 ranking is 0.857 on Places and 0.823 on ImageNet showing a notable correlation. See supplementary material for complete experimental details. 4.5. Universality of the Structure We employed a computational approach with various de- sign choices. It is important to investigate how specific to those the discovered structure is. We did stability tests by computing the variance in our output when making changes in one of the following system choices: I. architecture of 8
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  Transferring to ImageNet (Spearman’scorrelation = 0.823) Top-1 Top-5 Accuracy Transferring to MIT Places (Spearman’s correlation = 0.857) Top-1 Top-5 Accuracy Figure 12: Evaluating the discovered structure on other datasets: ImageNet [78] (left) for object classification and MIT Places [104] (right) for scene classification. Y-axis shows accuracy on the external benchmark while bars on x-axis are ordered by taxonomy’s predicted per- formance based on our dataset. A monotonically decreasing plot corre- sponds to preserving identical orders and perfect generalization. task-specific networks, II. architecture of transfer func- tion networks, III. amount of data available for training transfer networks, IV. datasets, V. data splits, VI. choice of dictionary. Overall, despite injecting large changes (e.g. varying the size of training data of transfer functions by 16x, size and architecture of task-specific networks and transfer networks by 4x), we found the outputs to be remarkably stable leading to almost no change in the output taxonomy computed on top. Detailed results and experimental setup of each tests are reported in the supplementary material. 4.6. Task Similarity Tree Thus far we showed the task space has a structure, mea- sured this structure, and presented its utility for transfer learning via devising transfer policies. This structure can be presented in other manners as well, e.g. via a metric of similarity across tasks. Figure 13 shows a similarity tree for the tasks in our dictionary. This is acquired from agglomer- ative clustering of the tasks based on their transferring-out behavior, i.e. using columns of normalized affinity matrix P as feature vectors for tasks. The tree shows how tasks would be hierarchically positioned w.r.t. to each other when measured based on providing information for solving other tasks; the closer two tasks, the more similar their role in transferring to other tasks. Notice that the 3D, 2D, low di- mensional geometric, and semantic tasks are found to clus- ter together using a fully computational approach, which matches the intuitive expectations from the structure of task space. The transfer taxonomies devised by BIP are consis- tent with this tree as BIP picks the sources in a way that all of these modes are quantitatively best covered, subject to the given budget and desired target set. 5. Limitations and Discussion We presented a method for modeling the space of visual tasks by way of transfer learning and showed its utility in reducing the need for supervision. The space of tasks is an interesting object of study in its own right and we have only Figure 13: Task Similarity Tree. Agglomerative clustering of tasks based on their transferring-out patterns (i.e. using columns of normalized affinity matrix as task features). 3D, 2D, low dimensional geometric, and semantic tasks clustered together using a fully computational approach. scratched the surface in this regard. We also made a number of assumptions in the framework which should be noted. Model Dependence: We used a computational approach and adopted neural networks as our function class. Though we validated the stability of the findings w.r.t various ar- chitectures and datasets, it should be noted that the results are in principle model and data specific. The current model also does not include a principled mechanism for handling uncertainty or probabilistic reasoning. Compositionality: We performed the modeling via a set of common human-defined visual tasks. It is natural to con- sider a further compositional approach in which such com- mon tasks are viewed as observed samples which are com- posed of computationally found latent (sub)tasks. Space Regularity: We performed modeling of a dense space via a sampled dictionary. Though we showed a good tolerance w.r.t. to the choice of dictionary and transferring to out-of-dictionary tasks, this outcome holds upon a proper sampling of the space as a function of its regularity. More formal studies on properties of the computed space is re- quired for this to be provably guaranteed for a general case. Transferring to Non-visual and Robotic Tasks: Given the structure of the space of visual tasks and demonstrated transferabilities to novel tasks, it is worthwhile to question how this can be employed to develop a perception module for solving downstream tasks which are not entirely visual, e.g. robotic manipulation, but entail solving a set of (a priori unknown) visual tasks. Lifelong Learning: We performed the modeling in one go. In many cases, e.g. lifelong learning, the system is evolving and the number of mastered tasks constantly increase. Such scenarios require augmentation of the structure with expansion mechanisms based on new beliefs. Acknowledgement: We acknowledge the support of NSF (DMS-1521608), MURI (1186514-1-TBCJE), ONR MURI (N00014-14-1-0671), Toyota(1191689-1-UDAWF), ONR MURI (N00014-13-1-0341), Nvidia, Tencent, a gift by Amazon Web Services, a Google Focused Research Award. 9
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