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Question: Meta-learning methods, most notably Model-Agnostic Meta-Learning or MAML, have achieved great success in adapting to new tasks quickly, after having been trained on similar tasks. The mechanism behind their success, however, is poorly understood. We begin this work with an experimental analysis of MAML, finding that deep models are crucial for its success, even given sets of simple tasks where a linear model would suffice on any individual task. Furthermore, on image-recognition tasks, we find that the early layers of MAML-trained models learn task-invariant features, while later layers are used for adaptation, providing further evidence that these models require greater capacity than is strictly necessary for their individual tasks. Following our findings, we propose a method which enables better use of model capacity at inference time by separating the adaptation aspect of meta-learning into parameters that are only used for adaptation but are not part of the forward model. We find that our approach enables more effective meta-learning in smaller models, which are suitably sized for the individual tasks. Meta-learning or learning to learn is an appealing notion due to its potential in addressing important challenges when applying machine learning to real-world problems. In particular, learning from prior tasks but being able to to adapt quickly to new tasks improves learning efficiency, model robustness, etc. A promising set of techiques, Model-Agnostic Meta-Learning or MAML, and its variants, have received a lot of interest (; ;). However, despite several efforts, understanding of how MAML works, either theoretically or in practice, has been lacking ). For a model that meta-learns, its parameters need to encode not only the common knowledge extracted from the tasks it has seen, which form a task-general inductive bias, but also the capability to adapt to new test tasks (similar to those it has seen) with task-specific knowledge. This begs the question: how are these two sets of capabilities represented in a single model and how do they work together? In the case of deep learning models, one natural hypothesis is that while knowledge is represented distributedly in parameters, they can be localized -for instance, lower layers encode task-general inductive bias and the higher layers encode adaptable task-specific inductive bias. This hypothesis is consistent with one of deep learning's advantages in learning representations (or feature extractors) using its bottom layers. Then we must ask, in order for a deep learning model to meta-learn, does it need more depth than it needs for solving the target tasks? In other words, is having a large capacity to encode knowledge that is unnecessary post-adaptation the price one has to pay in order to be adaptable? Is there a way to have a smaller (say, less deep) meta-learnable model which still adapts well? This question is of both scientific interest and practical importance -a smaller model has a smaller (memory) footprint, faster inference and consumes less resources. In this work, through empirical studies on both synthetic datasets and benchmarks used in the literature, we investigate these questions by analyzing how well different learning models can meta-learn and adapt. We choose to focus on MAML due to its popularity. Our observations suggest depth is indeed necessary for meta-learning, despite the tasks being solvable using a shallower model. Thus, applying MAML to shallower models does not in successful meta-learning models that can adapt well. Moreover, our studies also show that higher layers are responsible more for adapting to new tasks while the lower layers are responsible for learning task-general features. Our findings prompt us to propose a new method for meta-learning. The new approach introduces a meta-optimizer which learns to guide the (parameter) optimization process of a small model. The small model is used for solving the tasks while the optimizer bears the burden of extracting the knowledge of how to adapt. Empirical show that despite using smaller models, the proposed algorithm with small models attains similar performance to larger models which use MAML to meta-learn and adapt. We note that a recent and concurrent work to ours addresses questions in this line of inquiry . They reach similar through different analysis and likewise, they propose a different approach for improving MAML. We believe our work is complementary to theirs. Meta-learning, or learning-to-learn, is a vibrant research area with a long and rich history, lying at the intersection of psychology , neuroscience , and computer science Of particular interest to this manuscript is the line of work concerned with optimization-based meta-learning (OBML) algorithms in the few-shot regime, of which MAML is a particular instance.; ) Since its inception, MAML has been widely applied to tackle the few-shot learning challenge, in domains such as computer vision, natural language processing , and robotics . It is also the basis of extensions for continual learning , single-and multi-agent reinforcement learning , objective learning , transfer learning , and domain adaptation. Due to its generality, the adaptation procedure MAML introduceswhich is the focus of our analysis -has recently been branded as Generalized Inner Loop MetaLearning. While popular in practical applications, relatively few works have analysed the convergence and modelling properties of those algorithms. showed that, when combined with deep architectures, OBML is able to approximate arbitrary meta-learning schemes. recently provided convergence guarantees for MAML to approximate first-order stationary points for non-convex loss surfaces, under some assumptions on the availability and distribution of the data. Other analyses (empirical or theoretical) have attempted to explain the generalization ability of OBML , the bias induced by restricting the number of adaptation steps , or the effect of higher-order terms in the meta-gradient estimation Closely related to our proposed methods are works attempting to improve the adaptation mechanisms of OBML. Meta-SGD meta-learns per-parameter learning rates, while Alpha MAML adapts those learning rates during adaptation via gradient-descent. MetaCurvature propose to learn a Kronecker-factored pre-conditioning matrix to compute fast-adaptation updates. Their ing algorithm is a special case of one of our methods, where the linear transformation is not updated during adaptation. Another way of constructing preconditioning matrices is to explicitly decompose all weight matrices of the model in two separate components, as done in T-Nets . The first component is only updated via the evaluation loss, while the second is also updated during fast-adaptation. Warped Gradient Descent further extends T-Nets by allowing both components to be non-linear functions. Instead of directly working with gradients, suggest to directly learn a loss function which is differentiated during fast-adaptation and in faster and better learning. Additionally, meta-optimizers have also been used for meta-descent (; ; b; a;). They can be learned during a pre-training phase (; a; b; b; a; c;) or online (; ;). Our work differentiates from the the above by diagnosing and attempting to address the entanglement between modelling and adaptation in the meta-learning regime. We uncover a failure mode of MAML with linear and smaller models, and propose an effective solution in the form of expressive meta-optimizers. 3.1 In MAML and its many variants (; ;), we have a model whose parameters are denoted by θ. We would like to optimize θ such that the ing model can adapt to new and unseeen tasks fast. To this end, we are given a set of (meta)training tasks, indexed by τ. For each such task, we associate with a loss L τ (θ). Distinctively, MAML minimizes the expected task loss after an adaptation phase, consisting of a few steps of gradient descent from the model's current parameters. Since we do not have access to the target tasks to which we wish to adapt to, we use the expected loss over the training tasks, where the expectation is taken with respect to the distribution of the training tasks. α is the learning rate for the adaptation phase. The right-hand-side uses only one step gradient descent such that the aim is to adapt fast: in one step, we would like to reduce the loss as much as possible. In practice, a few more steps are often used. Shallow models can be hard to meta-learn Many intuitive explanations for why MAML works exist. One appealing suggestion is that the minimizer of the meta-learning loss L META is chosen in such a way that it provides a very good initialization for the adaptation phase; however, if the model is shallow such that L τ is convex in its parameters, then any initialization that is good for fast adapting to one subset of tasks could be bad for another subset of tasks since all the tasks have precisely one global minimizer and those minimizers can be arbitrarily far from each other. When the test tasks are distributed similar to the training tasks, the ideal initialization point has to be the "mean" of the minimizers of the training tasks -the precise definition of the mean is not important, as we will see below. We illustrate a surprising challenge by studying MAML on a synthetic dataset and the Omniglot task . Specifically, for the former study, we construct a set of binary classification tasks by first randomly sampling datapoints w ∈ R 100 from a standard Gaussian and use each of them to define a linear decision boundary of a binary classification task. We assume the boundaries pass through the origin and we sample training, validation and testing samples by randomly sampling data points from both sides of the decision boundaries. By construction, a linear model such as logistic regression is sufficient to achieve very high accuracy on any task. But can MAML learn a logistic regression model from a subset of training tasks that is able to adapt quickly to the test tasks? Note that due to the random sampling of the training tasks, the average of the minimizers (ie, the samples from the Gaussian distribution) is the origin. Likewise, for a set of test tasks randomly sampled the same way, the origin provides the best initialization by not favoring any particular task. Figure 1 reports the 1-step post-adaptation accuracy on the test tasks for the meta-learned logistic regression model. Surprisingly, the model fails to perform better than chance. Despite the simplicity of the tasks, logistic regression models are unable to find the origin as an initialization that adapts quickly to a set of test tasks that are similar to the training tasks. The models to be meta-learned are shallow ones (logistic regression models for binary classification and multinomial logistic regression) and deep ones -the shallow models with linear networks (LinNet) added to them. While the shallow models do not adapt and achieve chance level accuracies on average, the overparameterized versions attain significantly better after adaptation. Note that while the model has the same representational capacity as a linear logistic regression, it is overparameterized and has many local optimizers. As such, MAML can train this model such that its 1-step adaptation accuracy reaches 92% on average. We observe the same phenomena on meta-learning with MAML on the Omniglot dataset (details of the dataset are given in the Section 5). The shallow logistic regression model achieves 45% accuracy on average (for the 5-way classification tasks) after 2-step adaptation from the meta-learned initialization. However, with a linear network, the adapted model achieves significantly higher accuracy -70% on average, while having the same modeling capacity as the logistic regression. In summary, these experiments suggest that even for tasks that are solvable with shallow models, a model needs to have enough depth in order to be meta-learnable and to adapt. We postpone the description of our experiments on nonlinear models to section 5, where we also show that having sufficient depth is crucial for models to be meta-learnable, even when the tasks require fewer layers. A natural question arises: if being deep is so important for meta-learning on even very simple tasks, what different roles, if any, do different layers of a deep network play? Depth enables task-general feature learning and fast adaptation We hypothesize that for deep models meta-learned with MAML, lower layers learn task-invariant features while higher layers are responsible for fast-adaptation. To examine this claim, we meta-train a model consisting of four convolutional layers (C1 -C4) and a final fully-connected layer (FC) on Omniglot and CIFAR-FS . (Experimental setups are detailed in Section 5.) Once the model has finished meta-training, we perform a layer-wise ablation to study each layer's effect on adaptation. In particular, we iterate over each layer and perform two sets of experiments. In the first, we freeze the weights of the layer such that it does not get updated during fast-adaptation -we call it freezing only this. In the second experiment, we freeze all layers but the layer such that this layer is updated during fast-adaptation -we call it adapting only this. The left two plots in Figure 2 report the average accuracy over 100 testing tasks from both datasets. We observe that freezing only the first few lower layers (C1-C3) does not cause noticeable degradation to the post-adaptation accuracy. In fact, as long as the last convolutional layer (C4) is not frozen, post-adaptation accuracy remains unaffected. This indicate that C1-C3 provide information that is task-invariant, while C4 is crucial for adaptation. This does not mean FC is not important -since adapting C4 requires gradients passing through the FC layer, it cannot be arbitrary. In fact, in the rightmost plot of the figure, C1-C3 are held fixed during adaptation. While C4 and FC are allowed to adapt, and FC is perturbed with noise. When the noise is strong, the performance degrades significantly. Thus, we conclude that both C4 and FC play important roles in the mechanism for fast adaptation. We note that the recent work by concurently reached similar on the mini-ImageNet dataset, using feature similarity-based analyses of the model's representations. These observations highlight a fundamental issue: the property of being meta-learnable entails more model capacity than being learnable for a specific task. Thus, MAML can fail on models that lack the capacity to encode both task-general features and adaptation information, even when the models themselves are powerful enough to perform well on each of the tasks used for the meta-learning procedure. For example, with linear models (e.g. logistic regression), the parameters are forced to overlap and serve both modelling and adaptation purposes. However, as soon as the models are overparameterized, the extra layers enable meta-learnability. In section 5, we show that this observation also applies to nonlinear models where MAML-trained models quickly lose their performance when the number of layers is reduced. The previous section has shown that MAML, when applied to deep learning models, meta-learns both task-general features and task-specific adaption parameters at the same time. Since both are represented in a single model where strong dependencies exist between the lower layers and the higher layers, the model needs to be large enough and all its parameters have to be used to solve the test tasks. In some cases, the model has a modelling capacity that is bigger than what the test tasks require. For a model where linear network layers exist, the layers can be collapsed after adaptation so that the actual model used for inference on teh test tasks is small. However, it is not clear how to do so for typical deep models where nonlinearities prevent collapsing the model into smaller ones. Can we have a different adaptation mechanism such that a smaller model, whose modelling capacity is still adequate for the test tasks, can be adapted to find the minimizers of its loss on the aforementioned tasks? In this section, we describe a new approach of learning meta-optimizers for meta-learning. The meta-optimizer aims to separate modelling from adaptation. The meta-optimizer transforms the parameter update process for the model so the model can converge fast to its minimizer. How to transform is, however, learned from meta-training tasks. Because of this separation, the model for the task does not have to know how to adapt; it only needs to have enough capacity (roughly, big enough to represent the target task). We note that classical tools could be used to compress a large model for a given task. However, in the scenario where meta-learning is often used, such tools are unlikely effective. For example, distillation requires a lot of labeled data from the target task, which is not suitable for few-shot learning tasks. Pruning often degrades performance. A meta-optimizer -or learnable optimizer -is a parameterized function U ξ defining the model's parameter updates. For example, a linear meta-optimizer might be defined as: where ξ = (A, b) is the set of parameters of the linear transformation. The objective is to jointly learn the model and optimizer's parameters in the hope of accelerating the minimization of L τ. Concretely, we alternate between model and optimizer updates such that their values at adaptation step t + 1 are given by: While this change to the classical optimization pipeline might seem innocuous at first, it bears its own set of computational challenges. First, it is common for modern machine learning models to have millions of parameters. In this case, simply storing the optimizer's parameters becomes infeasible even for the simplest of models, such as the linear one outlined above. Because of this dimensionality issue most of the current literature considers a per-parameter independent update function, thus greatly limiting the expressivity of the meta-optimizer. We propose to address the issue of parameter dimensionality via factorizations of the optimizer's parameters, which we detail in the following subsection. To tackle the issue of parameter dimensionality we propose to factorize the optimizer's parameters. To demonstrate this, let us revisit the linear optimizer example above: we assume g ∈ R k to be the vectorization of a matrix gradient G ∈ R n×m, such that vec(G) = g and m · n = k. By expressing A = R ⊗ L as the Kronecker product of small matrices R ∈ R m×m and L ∈ R n×n, the linear update above can be efficiently computed as: (, Section 9.1) where b ∈ R k and ξ = (L, R, b). In the best case scenario where m = n = √ k, the above factorization requires O (k) memory as opposed to O k 2 for the non-factored case. Similarly, the matrix-product takes O k √ k time complexity, while the non-factored version takes O k 2. Note that this linear transformation can be used as a building block to implement more expressive and non-linear meta-optimizers. For example, a fully-connected network meta-optimizer U ξ is the composition of linear transformations U Wi interleaved with activation function σ. If the weight matrices (W 1, . . ., W h) of the network admit a Kronecker-factorization W i = R i ⊗ L i, the fullyconnected meta-optimizer is imlemented as: where ξ = (W 1, . . ., W h). Such a scheme can be used to implement arbitrary meta-optimizerssuch as convolutional or recurrent ones -so long as the architecture involves a composition of linear maps. We refer the reader to Appendix A.3 for pseudo-code and schematics of the model-optimizer loop. We emphasize that the choice of a Kronecker factorization is arbitrary; many matrix decompositions work equivalently well and in different modeling and computational trade-offs. For example, using a low-rank Cholesky factorization A = LL T where L ∈ R k×r allows to interpolate between computational complexity and decomposition rank by tuning the additional hyper-parameter r. The Cholesky decomposition might be preferable to the Kronecker one in memory-constrained applications, since r can be used to control the memory requirements of the meta-optimizer. Moreover, such a decomposition imposes symmetry and positiveness on A, which might be desirable when approximating the Hessian or its inverse. In this work, we prefered the Kronecker decomposition over alternatives for three reasons: the computational and memory cost of the Kronecker-product are acceptable, R ⊗ L is fullrank whenever L, R are full-rank, and the identity matrix lies in the span of Kronecker-factored matrices. In particular, this last motivation allows meta-optimizers to recover the gradient descent update by letting R, L be the identity. In our experiments, we complement our empirical studies and analysis from Section 3 with further and address the following question: can we meta-learn smaller (or shallower) models that perform as well as larger (or deeper) ones? To this end, we apply the proposed approach of learning meta-optimizers to the example synthetic dataset, as well as popular benchmark datasets: Omniglot , mini-ImageNet , and CIFAR-FS . All hyper-parameters and additional experimental details are available in Appendix A.1. Unless otherwise noted, all models are trained using MAML. For models that use learnable deep optimizers, we meta-learn both model and optimizer parameters. Similarly, both sets of parameters are adapted during fast-adaptation. To differentiate, we add the name of the optimizer to the model. For example, LR W/ KLIN corresponds to a logisitic regression model with the Kronecker-factorized linear transformation from the previous section. Similarly, W/ KFC indicates a model optimized by Kronecker-factored FullyConnected network in the previous section. We adopt the same setting as in section 3 where we study the synthetic data and the Omniglot dataset. For the Omniglot dataset, we consider the 1-shot, 5-way setup of and let the model adapt for 2 steps. Recall that the logistic regression model does not meta-learn while overparameterized logistic regression models (ie, models with linear networks) do. Table 1 reports the final post-adaptation testing accuracies and their standard deviations. As can be observed, while logistic regression (LR) cannot be meta-learned, overparameterized models (LINNET) as well as LR with our meta-optimizers (LR W/ KLIN and LR W/ KFC) can all metalearn while our approaches lead to similar or better compared to LinNet. Note that since linear networks can always be collapsed into a single weight matrix and bias term, any increase in performance is the consequence of better adaptation, rather than better modelling. We examine whether our meta-optimizers can meta-learn successfully with smaller but deep nonlinear models to match bigger deep models. We focus on two datasets: the Omniglot where the setting is 10-way classification with 5-shots and 4 adaptation steps, using the original 4-layer convolutional network (CNN) of , and the CIFAR-FS dataset , doing 10-way classification with 3-shots and 2 adaptation steps. The model is a CNN similar to the one for the Omniglot, but only uses 32 hidden units as opposed to 64. We report numerical in detail in Table A2 in the Appendix, which are graphically presented in Figure 3. We vary the number of convolutional layers. As the number of the layers decrease, the adaptation performance by both MAML and our approach (KFC) decrease. However, our approach has a much slower degradation rate. In fact, our approach is generally able to adapt a smaller model to the same level of performance as a MAML-trained model with an additional layer. We also compare our approach of learning meta-optimizers to other approaches, notably Meta-SGD and Meta-Curvature . Both of these methods attempt to approximate the expected task-loss landscape curvature and can be seen as ablations of our methods; Meta-Curvature corresponds to KLin without adapting the optimizer, while Meta-SGD approximates Meta-Curvature with a diagonal matrix. To ease the computational burden, we use a smaller CNN with 2 convolutional layers (SCNN) model for adaptation. As an upper bound on adaptation performance, we train the model on individual tasks to convergence and report the average final accuracy. Additionally, we include on the mini-ImageNet dataset for all methods, in the 10-way 1-shot with 5 adaptation steps setting . Table 2 reports post-adaptation testing accuracies. All methods of learning optimizers are able to adapt better than SCNN w/ MAML. In particular, our approachs perform the best and comes closest to approaching the upper bound of performance (SCNN). We introduce our approach by analyzing the success and failure modes of optimization-based metalearning methods. Namely, we find that, when successful, these methods tend to learn task-general features in early layers and adaptable parameters/update functions in the later layers. Moreover, we find that this learning fails when model size is reduced, indicating that optimization-based metalearning methods rely on the ability to encode task-general features and/or adaptable parameters, even when the model itself is adequate for learning on the individual tasks. As such, we introduce our method for decomposing modelling from adaptation using factored meta-optimizers. These meta-optimizers enable the forward model to use more capacity on learning task-specific features, while the expressiveness of their updates allows the forward model to adapt quickly to different tasks. We find that our approach is able to enable successful meta-learning in models that do not work with traditional optimization-based meta-learning methods. The following paragraphs describe the data-generation process, model architectures, and learning hyper-parameters for experiments in Section 5. Meta-optimizers's parameter matrices are always initialized to the identity. The task decision boundaries w ∼ N (0.0, I 100) are sampled from standard Gaussian. Each one of the 1,000 data point x ∼ N (0.0, I 100) is also sampled from the standard Gaussian, and is assigned a label from {−1, 1} based on the sign of w x. All models are trained for 30,000 iterations of stochastic gradient descent (without momentum) so as to minimize the 1-step MAML loss L MAML. The task loss L(θ) is the binary cross-entropy. Each stochastic gradient is estimated on a single task. (i.e. the meta-batch size is 1.) For the logistic regression (LR) , we used meta and adapatation learning rates of 0.01 and 0.5. The linear network (LR+LinNet) consists of 3 hidden layers of 64 units, and is trained with both meta-and fast-adaptation learning rates set to 1.9. Experiments using the Kronecker-factored linear meta-optimizer also use the same meta-and fast-adaptation learning rate, set to 3.33. The metaoptimizer itself consists of 10,001 parameters. (m = 100, n = 1) The Kronecker-factored fullyconnected meta-optimizer uses a single hidden layer with rectified linear-units (ReLU) activation functions, for a total of 20,002 parameters. (m = 100, n = 1) It has both learning rates set to 3.9. None of the meta-optimizers use a bias term. Omniglot Our Omniglot experiments exactly replicate the setup of. Of the 1623 classes, we designate 1,200 for meta-training, 100 for validation, and 423 for testing. We then generate 20,000 training tasks, 600 validation/testing tasks, each consisting of five classes containing a single character image, possibly rotated by 90, 180, or 270 degrees. All models are trained using Adam for 60,000 iterations, an iteration consisting of 32 tasks for which the model is allowed 2 steps of fast-adaptation. The logistic regression model uses a meta-learning rate of 0.0005 and an adaptation learning-rate set to 1.0. The linear network uses 4 hidden layers of 256, 128, 64, and 64 units and flattens the character images into a 784-dimensional vector before processing them. The meta-learning rate was set to 0.0005 and the adaptation learning rate to 0.08. For the KLin experiment (LR w/ KLin), we use a meta-learing rate of 0.003 and set the adaptation learning rate to 0.9. The KLin optimizer consists of 614k parameters, with m = 784, n = 5. The KFC meta-optimizer (LR w/ KFC) consists of 4 hidden layers (2.5M parameters) with ReLU activations, whose meta and adaptation learning rates are set to 0.001 and 0.01, respectively. Again, none of the meta-optimizer use biases. For the non-linear model experiments, we learn one KFC optimizer per layer of the model. Each KFC optimizer is based on the same architecture, but learns its own set of parameters. Moreover, we separate processing of the magnitude and direction of the model's gradient as follows: the gradient is initially scaled so as to have unit norm before being fed to the KFC network. Once the normalized gradient has been processed, it is rescaled by the initial normalizing constant times a learnable factor. (This learnable factor is initialized to 1.) We found this architecture to help in situations where the model is allowed several adaptation steps. For convolutional models, we flatten the height and width weight dimensions such that the optimizer is shared across filters of a same layer. (i.e. if the convolutional layer has shape NxCxHxW, we have m = C, n = HW .) Note that, again, none of the meta-optimizers use a bias term. Omniglot The dataset replicates the processing of linear model experiments, detailed above. Instead of 5 ways and 1 shot, we use 10 ways and 5 shots. All models are allowed 4 steps of fast adaptation. The 4-layer CNN network (CNN w/ MAML) uses a meta-learning rate of 0.003, and an adaptation learning rate of 0.5. Its architecture replicates the one described in. The 2-layer CNN (SCNN w/ MAML) uses the same first two layers and same input dimensionality to the FC layer as the 4-layer CNN, with meta and adapation learning rates set to 0.003 and 0.8, respectively. The 4-layer CNN has a total of 112,586 parameters, while the 2-layer only 38,474. We used the same learning rates for the Meta-SGD, Meta-Curvature, and KFC optimizers: a meta learning rate of 0.003, and an adaptation learning rate of 0.5. The KFC architecture consists of 4 layers, such that the meta-optimizers contains a total of 134,171 parameters for the 2-layer CNN model and 267,451 parameters for the 4-layer CNN. We obtained the splits created by and exactly reproduced their preprocessing setting for our experiments on CIFAR-FS. We also split the 100 classes in 64 training, 16 validation, and 20 testing classes and generate 20,000 training tasks, and 600 validation and testing tasks. We consider the 10-ways 3-shots setup with 2 fast-adaptation steps. As opposed to , our model closely resembles the one of our Omniglot experiments. Its main difference with the model described in prior work is the absence of max-pooling layers, and averaging of the last two convolutional feature dimensions before the fully-connected layer. Doing so allows us to conduct our ablation studies on the effect of our meta-optimizers with different number of convolutional layers, while keeping the input dimensionality of the fully-connected layer constant. The 4-and 2-layer CNNs use meta-and fast-adaptation learning rates of 0.003 and 0.7, respectively. The 4-layer CNN has a total of 29,226 parameters, while the 2-layer only 10,602. Meta-SGD, MetaCurvature, and KFC also use a 0.003 meta-learning rate, but the adaptation learning rate is decreased to 0.5. The KFC optimizer consists 4 layers with ReLU activations, for a total of 69,325 parameters for the 4-layer CNN model and 35,117 parameters for the 2-layer CNN. mini-ImageNet As for the Omniglot experiment, we replicate the mini-ImageNet setting and model from. We consider the 10-ways, 1-shot and 5 fast-adaptation steps setting. The 4-layer CNN and 2-layer CNN both use a meta-learning rate of 0.0005 and a fast-adaptation learning rate of 0.07. The 4-layer CNN has a total of 36,906 parameters, while the 2-layer only 18,282. The Meta-SGD, Meta-Curvature, and KFC meta-optimizer use the same meta-learning rate, but a smaller fast-adaptation learning rate set to 0.1 for Meta-SGD, 0.09 for Meta-Curvature, and 0.005 for KFC. For the 2-layer CNN model, the 4-layer KFC optimizer consists of approximately 2.60M parameters, while for the 4-layer CNN model it consists of 2.63M parameters. This section provides additional experimental evidence supporting the claims in the main text. Synthetic Figure A.1: Effect of width on adaptability on the synthetic binary classification setting. We vary the width and depth of linear networks and report their post-adaptation accuracy. As long as the model has one or more hidden layers, the model is able to adapt to the tasks regardless of the layers' width. In Section 3, we argue that depth is a curcial ingredient to enable gradient-based meta-learning. In particular, we show that even on simple, linearly separable binary classification tasks a logistic regression model (i.e. with no hidden layer) fails to solve the meta-learning problem. Using the same toy experiment, we now argue that width plays a relatively less important role with respect to adaptability. In Figure A.2, we plot the post-adaptation accuracy for different number of hidden layers, and vary their width w = 2, 4,..., 256. As can be seen, all models are able to eventually solve the metalearning problem regardless of width, so long as the model has one or more hidden layers. A.2.2 DO LAYERS C1-C3 BEHAVE SIMILARLY TO FC? FS (bottom). We observe that the modelling layers (C1-C3) are insensitive to parameter scaling regardless of whether it is applied pre-or post-adaptation, indicating they serve modelling purposes. On the other hand, parameter scaling on the adaptation layers (C4 & FC) quickly degrades postadaptation accuracy if applied pre-adaptation, but has virtually no effect if applied post-adaptation. In turn, this underlines the importance of these layers for adaptation. In Section 3, we claim that early layers in deep meta-trained models are purposed for modelling, while latter layers are responsible for fast-adaptation. This is reached through the ar-gument that (a) layers C1-C3 can be completely omitted from the adaptation dynamics, and (b) perturbing the last FC layer in large post-adaptation accuracy degradation. We now present further evidence that both layer groups (i.e. C1-C3 and C4-FC) indeed serve different purposes. For each layer of a meta-trained model, we multiply the weights of the selected layer by a constant factor either pre-or post-adaptation. Intuitively, layers that affect the fast-adaptation update will incur a large post-adaptation accuracy penalty as the scaling magnitude increases, due to the value of their weights playing an important part in the computation of the adaptation gradient. On the other hand, a constant scaling of the weights should have little effect on the modelling ability of the network, due to the relative magnitude of activation being maintained. (Note that this intuition is somewhat blurred for our models, as they make use of batch normalization.) Figure A.2 reports post-adaptation accuracy for those experiments. As predicted, scaling the weights of C1-C3 has little impact on accuracy, whether applied pre-or post-adaptation. Clearly, those layers do not affect the computation of the fast adaptation update. Similarly, scaling the weights of C4-FC post-adaptation does not impact the modelling ability of the network, and post-adaptation accuracy does not suffer. However, the same rescalings prove to be disastrous when applied pre-adaptation; the model is unable to adapt and accuracy drops by as much as 85% for Omniglot and 40% for CIFAR-FS. Evidently, those layers are central to successful fast-adaptation. In this section, we detail additional methods and apply them to the non-linear models of Section 5. Cholesky-factored Meta-Optimizers As alluded in Section 4, the choice of Kronecker decomposition is arbitrary. We now describe an alternative implementation of meta-optimizers, based on a low-rank Chholesky decomposition. The rank r Cholesky decomposition consists of a matrix L ∈ R n×r, such that a rank r symmetric positive definite matrix A ∈ R n×n is expressed as A = LL. Using this decomposition, we can write the linear transformation of Section 4 as: where b ∈ R n×1 is the bias term and g ∈ R n×1 is the vectorization of a weight's gradient. As for the Kronecker product, the latter expression of the linear transformation does not require to construct a large n × n matrix. Additionally, multiple linear transformations can be chained or interleaved by activation functions in order to obtain more expressive and adaptable meta-optimizers. A major attribute of the Cholesky decomposition is its flexibility in terms of memory requirements. For example, by letting r = 1 we force the meta-optimizer to have as many weights as the model. This attribute can in turn be a disadvantage: as we show in our experiments below, r plays an important role in the expressivity of the meta-optimizer, and should be treated as an additional hyper-parameter to tune. A second inconvenience stems from the choice of initialization scheme of the meta-optimizer; for the Kronecker product, we initialize the Kronecker factors to the identity, thus recovering hypergradient descent in the first meta-training batch. However, since the identity matrix I n does not lie in the span of low-rank Cholesky factorizable matrices, the initialization scheme becomes another decision left to the practitioner. For our subsequent experiments, we find that letting L follow a Glorot initialization (i.e. L ij ∼ N (0, gain · 1 n)) to work well. We compare the low-rank Cholesky-factored meta-optimizers to the Kronecker ones in Table A1. We denote by CFC1 the rank-1 Cholesky-factored fully-connected network, while CFC10 indicates the rank-10 version. CFC10 contains approximately as many parameters as KFC. (c.f. Table ? ?) While models trained with CFC do not outperform their KFC counter-parts, they offer a competitive alternative to existing methods. We study the effect of post-adaptation pruning of a meta-trained model. Ideally, pruning provides an attractive alternative to explicitly separating adaptation from modelling for learning lean models: practitioners could meta-train a larger model, adapt it to the desired target task, and finally prune it. We implement this pipeline by pruning weights smaller than a predefined threshold and report in Figure A. 3. As can be observed, pruning the 4-layer CNN models to obtain the equivalent number of parameters to a 2-layer SCNN can be catastrophic. On Omniglot, the pruned model reaches 70.06% post-adaptation accuracy, more than 20% lower than using metaoptimizers. On CIFAR-FS and mini-ImageNet, pruning can barely outperform chance prediction, reaching on average 10.00% and 12.20%, respectively. These are also available in Table A1, in the W/ PRUNE column. Sample task τ ∼ p(τ) 3: 4: for step t = 1,..., T do Compute loss L τ (θ t) Compute gradients ∇ θt L τ (θ t) and ∇ ξt L τ (θ t) Update the meta-optimizer parameters ξ t+1 = ξ t − α∇ ξt L τ (θ t) 8: Compute the model update U ξt+1 (∇ θt L τ (θ t)) 9: Update the model parameters θ t+1 = θ t − U ξt+1 (∇ θt L τ (θ t)) 10: end for 11: Update model and meta-optimizer initializations 12: 13: Table A3. All metrics are measured using a single NVIDIA Titan XP.

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We find that deep models are crucial for MAML to work and propose a method which enables effective meta-learning in smaller models.

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We introduce the NLC, a metric that is cheap to compute in the networks randomly initialized state and is highly predictive of generalization, at least in fully-connected networks.


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Question: Convolutional Neural Networks continuously advance the progress of 2D and 3D image and object classification. The steadfast usage of this algorithm requires constant evaluation and upgrading of foundational concepts to maintain progress. Network regularization techniques typically focus on convolutional layer operations, while leaving pooling layer operations without suitable options. We introduce Wavelet Pooling as another alternative to traditional neighborhood pooling. This method decomposes features into a second level decomposition, and discards the first-level subbands to reduce feature dimensions. This method addresses the overfitting problem encountered by max pooling, while reducing features in a more structurally compact manner than pooling via neighborhood regions. Experimental on four benchmark classification datasets demonstrate our proposed method outperforms or performs comparatively with methods like max, mean, mixed, and stochastic pooling. Convolutional Neural Networks (CNNs) have become the standard-bearer in image and object classification BID18. Due to the layer structures conforming to the shape of the inputs, CNNs consistently classify images, objects, videos, etc. at a higher accuracy rate than vector-based deep learning techniques BID18. The strength of this algorithm motivates researchers to constantly evaluate and upgrade foundational concepts to continue growth and progress. The key components of CNN, the convolutional layer and pooling layer, consistently undergo modifications and innovations to elevate accuracy and efficiency of CNNs beyond previous benchmarks. Pooling has roots in predecessors to CNN such as Neocognitron, which manual subsampling by the user occurs BID5, and Cresceptron, which introduces the first max pooling operation in deep learning BID28. Pooling subsamples the of the convolutional layers, gradually reducing spatial dimensions of the data throughout the network. The benefits of this operation are to reduce parameters, increase computational efficiency, and regulate overfitting BID1.Methods of pooling vary, with the most popular form being max pooling, and secondarily, average pooling BID18 BID13. These forms of pooling are deterministic, efficient, and simple, but have weaknesses hindering the potential for optimal network learning BID13 BID30. Other pooling operations, notably mixed pooling and stochastic pooling, use probabilistic approaches to correct some of the issues of the prior methods BID30 BID31.However, one commonality all these pooling operations employ a neighborhood approach to subsampling, reminiscent of nearest neighbor interpolation in image processing. Neighborhood interpolation techniques perform fast, with simplicity and efficiency, but introduce artifacts such as edge halos, blurring, and aliasing BID20. Minimizing discontinuities in the data are critical to aiding in network regularization, and increasing classification accuracy. We propose a wavelet pooling algorithm that uses a second-level wavelet decomposition to subsample features. Our approach forgoes the nearest neighbor interpolation method in favor of an organic, subband method that more accurately represents the feature contents with less artifacts. We compare our proposed pooling method to max, mean, mixed, and stochastic pooling to verify its validity, and ability to produce near equal or superior . We test these methods on benchmark image classification datasets such as Mixed National Institute of Standards and Technology (MNIST) BID12, Canadian Institute for Advanced Research (CIFAR-10) BID11, Street House View Numbers (SHVN) BID17, and Karolinska Directed Emotional Faces (KDEF) . We perform all simulations in MATLAB R2016b. The rest of this paper organizes as follows: Section 2 gives the , Section 3 describes the proposed methods, Section 4 discusses the experimental , and Section 5 gives the summary and . Pooling is another term for subsampling. In this layer, the dimensions of the output of the convolutional layer are condensed. The dimensionality reduction happens by summarizing a region into one neuron value, and this occurs until all neurons have been affected. The two most popular forms of pooling are max pooling and average pooling BID18 BID13. Max pooling involves taking the maximum value of a region R ij and selecting it for the condensed feature map. Average pooling involves calculating the average value of a region and selecting it for the condensed feature map. The max pooling function is expressed as: DISPLAYFORM0 While average pooling is shown by the following equation: DISPLAYFORM1 Where a kij is the output activation of the k th feature map at (i,j), a kpq is the input activation at (p,q) within R ij, and |R ij | is the size of the pooling region. An illustration of both of these pooling methods is expressed in FIG0 BID29: While max and average pooling both are effective, simple methods, they also have shortcomings. Max pooling, depending on the data, can erase details from an image BID30 BID31. This happens if the main details have less intensity than the insignificant details. In addition, max pooling commonly overfits training data BID30 BID31. Average pooling, depending on the data, can dilute pertinent details from an image. The averaging of data with values much lower than significant details causes this action BID30 BID31. FIG1 illustrates these shortcomings using the toy image example:To combat these issues, researchers have created probabilistic pooling methods. Mixed pooling combines max and average pooling by randomly selecting one method over the other during training BID30. There is no set way to perform mixed pooling. This method is applied arbitrarily in three different ways for all features within a layer, mixed between features within a layer, or mixed between regions for different features within a layer BID13 BID30. Mixed pooling is shown in the following equation: DISPLAYFORM2 where λ is a random value 0 or 1, indicating max or average pooling for a particular region/feature/layer. Another probabilistic pooling method, called stochastic pooling, improves upon max pooling by randomly sampling from neighborhood regions based on the probability values of each activation BID31. These probabilities p for each region are calculated by normalizing the activations within the region: DISPLAYFORM3 The pooled activation is sampled from a multinomial distribution based on p to pick a location l within the region BID31. The process is captured in the following equation BID31:. In any given region, the activations with the highest probabilities have the higher chance of selection. However, any activation can be chosen. In this example, the stochastic pooling method selects the midrange activation with a probability of 13%. By being based off probability, and not deterministic, stochastic pooling avoids the shortcomings of max and average pooling, while enjoying some of the advantages of max pooling BID31. DISPLAYFORM4 The previously highlighted pooling methods use neighborhoods to subsample, almost identical to nearest neighbor interpolation. Previous studies explore the possibilities of wavelets in image interpolation versus traditional methods BID4. Our proposed pooling method uses wavelets to reduce the dimensions of the feature maps. We propose using the wavelet transform to minimize artifacts ing from neighborhood reduction BID20. We postulate that our approach, which discards the first-order subbands, more organically captures the data compression. This organic reduction therefore lessens the creation of jagged edges and other artifacts that may impede correct image classification. The proposed wavelet pooling scheme pools features by performing a 2nd order decomposition in the wavelet domain according to the fast wavelet transform (FWT) BID15 BID16 BID23 BID2, which is a more efficient implementation of the two-dimensional discrete wavelet transform (DWT) as follows BID3 BID24 BID21: DISPLAYFORM0 DISPLAYFORM1 where ϕ is the approximation function, and ψ is the detail function, W ϕ, W ψ are called approximation and detail coefficients. h ϕ [−n] and h ψ [−n] are the time reversed scaling and wavelet vectors, (n) represents the sample in the vector, while (j) denotes the resolution level. When using the FWT on images, we apply it twice (once on the rows, then again on the columns). By doing this in combination, we obtain our detail subbands (LH, HL, HH) at each decomposition level, and our approximation subband (LL) for the highest decomposition level. After performing the 2nd order decomposition, we reconstruct the image features, but only using the 2nd order wavelet subbands. This method pools the image features by a factor of 2 using the inverse FWT (IFWT) BID15 BID16 BID23 BID2, which is based off of the inverse DWT (IDWT) BID3 BID24 BID21: FIG3 gives an illustration of the algorithm for the forward propagation of wavelet pooling: The proposed wavelet pooling algorithm performs backpropagation by reversing the process of its forward propagation. First, the image feature being back propagated undergoes 1 st order wavelet decomposition. After decomposition, the detail coefficient subbands upsample by a factor of 2 to create a new 1 st level decomposition. The initial decomposition then becomes the 2 nd level decomposition. Finally, this new 2 nd order wavelet decomposition reconstructs the image feature for further backpropagation using the IDWT. FIG4 details the backpropagation algorithm of wavelet pooling: DISPLAYFORM2 All CNN experiments use MatConvNet BID26. All training uses stochastic gradient descent BID0. For our proposed method, the wavelet basis is the Haar wavelet, mainly for its even, square subbands. All experiments are run on a 64-bit operating system, with an Intel Core i7-6800k CPU @ 3.40 GHz processor, with 64.0 GB of RAM. We utilize two GeForce Titan X Pascal GPUs with 12 GB of video memory for all training. All CNN structures except for MNIST use a network loosely based on Zeilers network BID31. We repeat the experiments with Dropout BID22 and replace Local Response Normalization BID11 with Batch Normalization BID7 for CIFAR-10 and SHVN (Dropout only) to examine how these regularization techniques change the pooling . To test the effectiveness of each pooling method on each dataset, we solely pool with that method for all pooling layers in that network. All pooling methods use a 2x2 window for an even comparison to the proposed method. Figure 6 gives a selection of each of the datasets. The network architecture is based on the example MNIST structure from MatConvNet, with batch normalization inserted. All other parameters are the same. The input training data and test data come from the MNIST database of handwritten digits. The full training set of 60,000 images is used, as well as the full testing set of 10,000 images. TAB0 shows our proposed method outperforms all methods. Given the small number of epochs, max pooling is the only method to start to overfit the data during training. Mixed and stochastic pooling show a rocky trajectory, but do not overfit. Average and wavelet pooling show a smoother descent in learning and error reduction. FIG6 shows the energy of each method per epoch. We run two sets of experiments with the pooling methods. The first is a regular network structure with no dropout layers. We use this network to observe each pooling method without extra regularization. The second uses dropout and batch normalization, and performs over 30 more epochs to observe the effects of these changes. FIG7 shows our network structure for the CIFAR-10 experiments:The input training and test data come from the CIFAR-10 dataset. The full training set of 50,000 images is used, as well as the full testing set of 10,000 images. For both cases, with no dropout, and with dropout, TAB1 show our proposed method has the second highest accuracy. Max pooling overfits fairly quickly, while wavelet pooling resists overfitting. The change in learning rate prevents our method from overfitting, and it continues to show a slower propensity for learning. Mixed and stochastic pooling maintain a consistent progression of learning, and their validation sets We run two sets of experiments with the pooling methods. The first is a regular network structure with no dropout layers. We use this network to observe each pooling method without extra regularization. The second uses dropout to observe the effects of this change. FIG0 shows our network structure for the SHVN experiments:The input training and test data come from the SHVN dataset. For the case with no dropout, we use 55,000 images from the training set. For the case with dropout, we use the full training set of 73,257 images, a validation set of 30,000 images we extract from the extra training set of 531,131 images, as well as the full testing set of 26,032 images. For both cases, with no dropout, and with dropout, FIG0: CNN SHVN Structure Block Diagram TAB3 and TAB4 show our proposed method has the second lowest accuracy. Max and wavelet pooling both slightly overfit the data. Our method follows the path of max pooling, but performs slightly better in maintaining some stability. Mixed, stochastic, and average pooling maintain a slow progression of learning, and their validation sets trend at near identical rates. FIG0 shows the energy of each method per epoch. We run one set of experiments with the pooling methods that includes dropout. FIG0 shows our network structure for the KDEF experiments:The input training and test data come from the KDEF dataset. This dataset contains 4,900 images of 35 people displaying seven basic emotions (afraid, angry, disgusted, happy, neutral, sad, and surprised) using facial expressions. They display emotions at five poses (full left and right profiles, half left and right profiles, and straight). This dataset contains a few errors that we fix (missing or corrupted images, uncropped images, etc.). All of the missing images are at angles of -90, -45, 45, or 90 degrees. We fix the missing and corrupt images by mirroring their counterparts in MATLAB and adding them back to the dataset. We manually crop the images that need to match the dimensions set by the creators (762 x 562). KDEF does not designate a training or test data set. We shuffle the data and separate 3,900 images as training data, and 1,000 images as test data. We resize the images to 128x128 because of memory and time constraints. The dropout layers regulate the network and maintain stability in spite of some pooling methods known to overfit. TAB5 shows our proposed method has the second highest accuracy. Max pooling eventually overfits, while wavelet pooling resists overfitting. Average and mixed pooling resist overfitting, but are unstable for most of the learning. Stochastic pooling maintains a consistent progression of learning. Wavelet pooling also follows a smoother, consistent progression of learning. FIG0 shows the energy of each method per epoch. TAB5: KDEF Performance of Pooling Methods + Dropout 4.5 COMPUTATIONAL COMPLEXITY Our construction and implementation of wavelet pooling is not efficient. We present this proposed methods as a proof-of-concept, to show its potential and validity, and also to be open to massive improvements. The main area of improvement is computational efficiency. As a proof-of-concept, the code written to implement this method is not at its peak form. Additionally, we did not have the time, space, or resources to optimize the code. We view the accuracy and novelty as a starting point to spawn improvements, both from our own research as well as other researchers. We calculate efficiency in terms of mathematical operations (multiplications, additions, logical, etc.) that each method utilizes to complete its algorithm. For max pooling, we calculate operations based on the worst-case scenarios for each neighborhood in finding the maximum value. For average pooling, we calculate the number of additions and division for each neighborhood. Mixed pooling is the mean value of both average and max pooling. We calculate operations for stochastic pooling by counting the number of mathematical operations as well as the random selection of the values based on probability (Roulette Wheel Selection). For wavelet pooling, we calculate the number of operations for each subband at each level, in both decomposition and reconstruction. TAB7 shows the number of mathematical operations for one image in forward propagation. This table shows that for all methods, average pooling has the least number of computations, followed by mixed pooling, with max pooling not far behind. Stochastic pooling is the least computationally efficient pooling method out of the neighborhood-based methods. It uses about 3x more mathematical operations than average pooling, the most computationally efficient. However, wavelet pooling by far is the least computationally efficient method, using 54 to 213x more mathematical operations than average pooling. This is partially due to the implementation of the subband coding, which did not implement multidimensional decomposition and reconstruction. MNIST Nonetheless, by implementing our method through good coding practices (vectorization, architecture, etc.), GPUs, and an improved FTW algorithm, this method can prove to be a viable option. There exists a few improvements to the FTW algorithm that utilize multidimensional wavelets BID8 BID27, lifting BID25, parallelization BID6, as well as other methods that boast of improving the efficiency in speed and memory BID19 BID9 BID10 We prove wavelet pooling has potential to equal or eclipse some of the traditional methods currently utilized in CNNs. Our proposed method outperforms all others in the MNIST dataset, outperforms all but one in the CIFAR-10 and KDEF datasets, and performs within respectable ranges of the pooling methods that outdo it in the SHVN dataset. The addition of dropout and batch normalization show our proposed methods response to network regularization. Like the non-dropout cases, it outperforms all but one in both the CIFAR-10 & KDEF datasets, and performs within respectable ranges of the pooling methods that outdo it in the SHVN dataset. Our confirm previous studies proving that no one pooling method is superior, but some perform better than others depending on the dataset and network structure BID1; BID13. Furthermore, many networks alternate between different pooling methods to maximize the effectiveness of each method. Future work and improvements in this area could be to vary the wavelet basis to explore which basis performs best for the pooling. Altering the upsampling and downsampling factors in the decomposition and reconstruction can lead to better image feature reductions outside of the 2x2 scale. Retention of the subbands we discard for the backpropagation could lead to higher accuracies and fewer errors. Improving the method of FTW we use could greatly increase computational efficiency. Finally, analyzing the structural similarity (SSIM) of wavelet pooling versus other methods could further prove the vitality of using our approach.

Pooling is achieved using wavelets instead of traditional neighborhood approaches (max, average, etc).

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We propose a new compression method, Inter-Layer Weight Prediction (ILWP) and quantization method which quantize the predicted residuals between the weights in convolution layers.


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Question: Dynamic ridesharing services (DRS) play a major role in improving the efficiency of urban transportation. User satisfaction in dynamic ridesharing is determined by multiple factors such as travel time, cost, and social compatibility with co-passengers. Existing DRS optimize profit by maximizing the operational value for service providers or minimize travel time for users but they neglect the social experience of riders, which significantly influences the total value of the service to users. We propose DROPS, a dynamic ridesharing framework that factors the riders' social preferences in the matching process so as to improve the quality of the trips formed. Scheduling trips for users is a multi-objective optimization that aims to maximize the operational value for the service provider, while simultaneously maximizing the value of the trip for the users. The user value is estimated based on compatibility between co-passengers and the ride time. We then present a real-time matching algorithm for trip formation. Finally, we evaluate our approach empirically using real-world taxi trips data, and a population model including social preferences based on user surveys. The demonstrate improvement in riders' social compatibility, without significantly affecting the vehicle miles for the service provider and travel time for users. Dynamic ridesharing services, such as UberPool and LyftLine, are becoming an increasingly popular means of commute, especially in large cities BID6 BID2. Dynamic ridesharing is characterized by matching multiple requests that arrive in real-time, for a one-way and one-time trip. We consider a setting in which a service provider operates a vehicle fleet and schedules cars to pick up and drop off passengers in response to a stream of requests, which includes matching requests with each other. There are two important factors that explain the growing attractiveness of DRS for customers: (i) cost effectiveness and (ii) ease of finding a ride in large cities where it is comparatively hard to find a taxi otherwise. For a service provider, dynamic ridesharing helps serve customers with possibly fewer vehicles, thus reducing their operational cost. A common objective for optimizing riders' satisfaction in existing ridesharing systems is to minimize travel time BID14 BID0 BID2. In practice, however, there are many other factors that affect user satisfaction in dynamic ridesharing, apart from travel time. Since a user could be traveling with strangers in the ride, their compatibility plays a major role in the user's satisfaction. In fact, there is growing evidence that desire for personal space and security when riding with strangers pose a major barrier to using ridesharing for many users (Tao and Wu 2008; BID0 . For example, a female passenger may prefer to ride only with female co-passengers. The user may have a different set of preferences depending on the time of day and the location -preferences are tripspecific and not necessarily user-specific. Consider a scenario with three requests where r 1 and r 2 are male and r 3 is a female passenger. Let these requests arrive at the same time FIG0), such that optimizing the operational value for the service provider forms a trip with these requests (1(a)). However, this may violate the users' social preferences and the trip may need to be altered to satisfy the preferences, such as the following:• If the passengers prefer riding with co-passengers of the same gender but are indifferent to riding with copassengers of a different gender, then it is desirable to minimize their ride time overlap in the vehicle by altering the pick up and drop off order FIG0 ); and If the service does not provide a mechanism to express such social preferences and forms trips that violate these preferences (as in 1(a)), the customers may not use the service. Current DRS, however, do not account for social preferences in their optimization, despite being indicated as a major concern for users in several surveys BID0 BID15 BID10 BID21. We present DROPS (Dynamic Ridesharing Optimization using social PreferenceS), a dynamic ridesharing framework that facilitates incorporating social preferences of the users in the trip formation process. A weight vector over preferences indicates the importance of each factor in determining the trip value to the user. The goal is to form trips that optimize both operational value for the service provider and value of the trip to the passengers, which incentivizes them to continue using the service and benefits the service provider. The value of a trip to a user is calculated based on their social compatibility with other co-passengers, the ride time, and ride cost. We solve this bi-objective optimization problem using scalarization BID18, which solves a linear combination of the multiple objectives. The relative importance of each objective can be controlled using the weight vector for the objectives. Given a set of riders, we evaluate their potential shared trip using an optimal trajectory planning algorithm. Candidate trips are formed using our real-time greedy algorithm that adds customers to a trip only if the trip's value is above a certain threshold. We consider three basic social factors -age, gender, and user rating-along with a time preference indicating if the user is in a rush. The viability of factoring social preferences into the trips scheduling process is evaluated empirically. The experiments examine the impact of matching with social preferences (social matching) on users and the service provider. We test our approach on a real-world taxi trips dataset and compare the with that of three baselines, each focusing on optimizing different components of the objective for trip formation. The population model and preferences used in our experiments were acquired using webbased user surveys, which was conducted in two phases and had 489 responses. The survey was conducted specifically to determine how different potential riders evaluate social ridesharing. Our show that incorporating social preferences of users in the trip formation improves the overall user satisfaction, without significantly affecting the operational cost for the service provider. Our primary contributions are: (i) presenting DROPS, a system for dynamic ridesharing with social preferences; (ii) proposing a real-time greedy algorithm for trip formation; and (iii) extensive empirical evaluation showing the benefits of social matching in dynamic ridesharing using real-world taxi data and a population model based on user surveys. Dynamic ridesharing has gained popularity since the early 2000's due to the cost benefits it offers to the users and service providers, apart from its contributions to sustainable environment ing from efficient vehicle usage. Dynamic ridesharing is characterized by user requests that arrive in real-time and are matched with vehicles BID13. Another popular ridesharing setting is the car-pooling where users travel together for a particular purpose and the trips are usually recurring BID6. Our work differs from car-pooling as we focus on a dynamic ridesharing setting with a service provider who operates the vehicle fleet instead of individual car owners and trips that are typically non-recurring. Optimizing dynamic ridesharing services has been an active research area, attracting researchers from diverse fields such as operations research, transportation, and artificial intelligence BID0 BID6 BID8 BID1. Existing literature on dynamic ridesharing can be classified broadly based on the objective function and the solution method employed. Optimization-based approaches are the common solution technique employed BID19 BID14 BID8 BID5 AlonsoMora et al. 2017; BID9 BID3. Other approaches include partition-based BID17, auction-based mechanisms BID7, and genetic algorithms BID11. Researchers have employed these techniques largely to optimize the routing or travel time BID10 BID0 BID11 BID17 BID19 BID5. Specifically, the commonly used objectives for determining ridesharing matches are: (i) minimizing system-wide vehicle-miles; (ii) minimizing system-wide travel time; and (iii) maximizing number of participants. A critical missing component of these objectives is the in-ride user experience. Numerous studies have outlined the need for learning and understanding user preferences in the context of ridesharing, beyond simple factors like time windows BID6 BID0 BID22. Multiple surveys have acknowledged that it is essential to account for users' social preferences to improve dynamic ridesharing BID0 BID8 BID10 BID20 BID6 BID17 Tao and Wu 2008; BID16 BID21 BID4. To address this discrepancy, we present a dynamic ridesharing framework that allows for representing and satisfying the social preferences of the users in trip formation. The DROPS framework facilitates customizing rides to improve user compatibility by incorporating the social preferences of users. Let R t denote the finite set of unserved (nondispatched) requests at time t and V t denote the finite set of available vehicles at time t. Each request r ∈ R t is denoted by s, e, i, p, U. Each vehicle v ∈ V t is denoted by the tuple ID, ω. Refer Table 1 for the definitions of variables and constants employed in the formulation. We consider social preferences in each request that correspond to three social factors: age, gender, and rating of users. Additionally, we consider a time preference to indicate if the user is in a rush. We identified these factors based on the of our user surveys, conducted specifically to determine user expectations in ridesharing services. The preferences (p) are denoted as +1, −1, or 0, indicating the user's desirability, undesirability, or indifference about a certain value of a factor. For example, a preference of +1 for rating ≥ 4 denotes that the person prefers riding with co-passengers who have a minimum rating of 4, and a preference of −1 for rating ≤ 3 denotes that the person wishes to avoid riding with co-passengers who have a rating of 3 or below. That is, if rating on a scale of 1 to 5 is treated as a vector, then these preferences are denoted as −1, −1, −1, +1, +1. The weights w = [w t, w a, w g, w s]T correspond to the time, age, gender, and rating, respectively. A solution to an instance of this problem is a set of trips Λ, where each trip λ ∈ Λ is a matching of requests to a vehicle and is denoted by R, v, τ. The value of a trip is denoted by V (λ). The objective is to maximize the cumulative value of all trips dispatched in a given horizon H, DISPLAYFORM0 Multi-objective formulation Since the goal is to schedule trips that maximize the operational value for the service provider as well as maximizing the overall user value, this is naturally a bi-objective optimization. To solve this, we employ scalarization BID18, which projects a multi-objective value to a single scalar value by parameterizing the objectives using a weight vector. The weight value for each objective indicates its relative importance, thus ing in a single objective function for optimization. Let β o denote the weight corresponding to the operational value and let β u denote the weight corresponding to the user value. Then, ∀λ, the trip value is: DISPLAYFORM1 The operational value and the user value are measured in dollars ($) and normalized to the same scale before scalarization. The operational value to the service provider depends on the cost of operating the vehicle for the trip c τ λ and the amount paid by the riders, which is the difference between the amount charged for the trip (x r) and the discount offered for using ridesharing (d r). The value of the trip to a user depends on the user utility (α r) and the discount gained for using ridesharing (d r). The user utility (α r) is the difference between the users' social compatibility with all their co-passengers and the extra travel time incurred by using ridesharing. The social compatibility for a request is calculated as the cumulative weighted difference between the preferences p r and demographics of each co-passenger. We now explain the social utility calculation using a simple example. Consider two requests r 1 (female) and r 2 (male) that arrive at the same time and have the same source and destination coordinates, same age, and rating. of r for the trip α r User's social utility x r Amount charged to r for the trip d r Discount for using ridesharing i r Request initiation time p r Social and time preferences of r w r User's weights for preferences p r U r User demographics: {age, gender, rating} ID v Vehicle ID Given a set of requests and vehicles, our solution approach consists of two components: (i) trip formation and (ii) trip dispatch. FIG2 is an illustration of our solution approach. In each decision cycle, the trip formation component matches requests with each other and to vehicles, and the dispatch component decides which trips are dispatched. We restrict the scope of matching in this paper to requests and vehicles that have not been dispatched. That is, we do not consider a vehicle en-route (already driving on the road) in the scheduling process and therefore do not match requests to such vehicles. The route planner calculates the optimal trajectory for picking up and dropping off a given set of requests. In this phase, requests are matched with other requests and assigned a vehicle to form a trip. The matching is performed using a greedy approach outlined in Algorithm 1. The input to the algorithm is the set of requests and a trip value threshold δ that indicates the required minimum improvement in trip value to form trips. The algorithm adds a request to the best trip (maximum improvement) that improves the trip value at least by a factor of δ and if the trip size has not exceeded the maximum capacity of the vehicle (Lines 7-16). Standard hyperparameter tuning or sample average Each request is assigned to the best trip that satisfies the threshold improvement. If no such trip is found, then a new trip is created with the request. This ensures that all requests are associated with a trip. The route planner computes trajectories that determine the pick up and drop off order for a given set of requests. All possible trajectories are generated and the one that maximizes the trip value for a given set of requests is selected as the route τ for the trip. During the trip formation, the best route is updated whenever a new request is added to a trip (Line 8, 21). The output of this algorithm is the set of all trips formed, Λ t.Partitioning Requests for Scalability The computational complexity of the matching algorithm discussed above increases rapidly with the increase in number of requests. To counter this computational complexity, we exploit the notion of independence among requests. Two requests q and r are said to be independent if serving them together in the same trip is not desirable in terms of trip value. Hence, all the requests over different days or requests with non-overlapping source-destination pairs are independent. The requests can be partitioned based on their dependence and matches may be formed among each partition in parallel. Sometimes, it is non-trivial to estimate an exact partitioning of requests with respect to routes, without forming trips and calculating the best route possible. In such cases, the underlying map may be partitioned into geographic zones to form trips in each zone independently by considering the requests originating in that zone, as in our experiments. The trips formed in the matching phase are dispatched in this phase if at least one of the following conditions is satisfied: (i) trip value is above the predefined dispatch threshold; or (ii) a request in the trip has remained unserved for a certain period of time since its arrival (queue time). The dispatch threshold for trip value and the queue time for the requests are determined by the service provider. For example, all requests that are unserved for five minutes or more since their arrival time may be dispatched irrespective of the trip value, depending on vehicle availability. In our experiments, trips Algorithm 1: Greedy Matching (R t, δ) DISPLAYFORM0 Calculate best route for λ = λ + r that satisfy the queue time threshold are given a higher priority over the trips with lower queue time but higher trip value. This ensures that certain requests do not remain unserved forever due to lower trip value. The trips are then dispatched based on availability of vehicles, V t. At the end of decision cycle t, all unserved requests -requests in trips that are not dispatched -are added to the requests set for the next decision cycle, R t+1. DISPLAYFORM1 The experiments evaluate the impact of using social preferences in ridesharing, with respect to users and the service provider. We built a realistic simulator of ridesharing using the Chicago taxi trips dataset 1 and a population model based on extensive user surveys. We compare the obtained using social preferences in dynamic ridesharing matching (SM) with that of three baselines: (B 1) maximizing only the operational value, β u = 0, β o = 1; (B 2) maximizing only user value, β u = 1, β o = 0; and (B 3) maximizing the comprehensive trip value in Equation 1 but without considering user's social preferences corresponding to age, gender, and rating, w a = 0, w g = 0, w s = 0, for the trip formation. Algorithm 1 is used to form trips using each baseline objective. The algorithms and the simulation system were implemented by us on an Intel Xeon 3.10 GHz computer with 16GB of RAM, using a homogeneous vehicle fleet with a 1 https://data.cityofchicago.org/Transportation/TaxiTrips/wrvz-psew seat capacity of 4 for the evaluation. Each decision cycle is 30 seconds in real-time and the horizon H is one day. We assume that the number of vehicles is not bounded since the benefits of social matching are best illustrated in this case and all techniques are equally affected by vehicle restriction. We set the trip threshold δ to zero for the greedy algorithm; requests are added to the best trips possible as long as the current value of the trip is not diminished. This allows us to examine the benefit of social matching uniformly across zones by using a conservative value. However, in practice this hyperparameter may be tuned to further optimize performance subject to the service provider's objective. The request queue time threshold for dispatch is set to five minutes. The travel time and distances are calculated using straight line distances between the coordinates and a vehicle speed of 30 miles per hour. While these experiments do not account for the actual routes and traffic conditions, these factors are not likely to change the relative merits of each approach and the of the study. The population model considered in our experiments is based on the of online surveys that was conducted in North America. The survey had 489 responses which indicated that users would like to be matched with people who are similar to them. The demographic information such as age and gender, for our experiments, is drawn from the actual Chicago demographic distributions 2. The preferences (p) and the weights (w) are based on the survey . The survey also indicates that some users are unwilling to use ridesharing when social preferences are not taken into account. To reflect this, certain users were marked as reluctant for ridesharing in the absence of social matching and these users were always dispatched as solo rides, when forming trips with the baseline objectives. The Chicago taxi trips data consists of trip-specific information such as start time and end time of the taxi ride, trip duration, trip fare, and the latitude and longitude coordinates for pick up and drop off locations along with the geographic zone corresponding to these locations. A map of Chicago divided into zones 3 is shown in FIG4. We partition the data from each zone into training and testing sets. The weights for scalarization were estimated empirically using the training data FIG3 ). In FIG3, the x-axis is the weight for operational value (β o) and the y-axis denotes the weight corresponding to user value (β u). The weights used for the test sets are β o = 0.8 and β u = 0.6 for zones 8 and 28, and we used β o = 0.5 and β u = 0.5 for experiments on zone 56. Our algorithm is evaluated along different metrics on the test set which uses data from two consecutive weeks in April 2015. We consider requests originating in zones 8, 28, and 56, whose requests densities are high, medium, and low respectively. The average number of requests per day in each of these zones is 20000, 7000, and 1500 respectively. Since user value and the operational value are often competing metrics, we analyze the quality of trips formed with respect to each of these. We measure the impact on users based on the total user value (Figure 5), average social utility per minute (Figure 6), and the increase in ride time, relative to a solo trip FIG6.Trips formed by maximizing operational value (B 1) have the least user value across all zones, as expected. Our approach (SM) achieves user value close to that of optimiz- ing for user value alone (B 2), and sometimes better than B 2. This is because, in some cases, the values of the trips formed by optimizing B 2 objective may not meet the dispatch threshold in which the case the trips are dispatched after five minutes, which eventually reduces the user value. Our approach overcomes this drawback by optimizing for both the objectives, providing greater cumulative value for a given trip and enabling it to be dispatched more quickly. The social utility (α r) per minute measures the average social compatibility of users with their co-passengers. To account for the different ride times of the trips, we measure the average utility per minute, along with standard error (Figure 6). We observe that SM consistently performs similar to or better than B 2, showing that the user value is improved through better matching, and not merely based on the ride time or discount offered. We also evaluated the increase in ride time of the different techniques, compared to solo ride FIG6. The average ride times are in the range 10-20 minutes for requests originating in zones of interest. Though the increase in ride time of our technique is around three minutes, note that ridesharing, in general, incurs additional ride time. The increase in ride time of our technique is well within the range that users consider acceptable (at most 5 minutes) according to the survey . The social compatibility typically offsets the increase in ride time for the users, thus ing in increased user utility when forming trips using our approach. Impact on the Service Provider The impact on service provider is determined based on the operational value and the total miles driven, to give a sense of degree of variation induced by social matching on the trip routes and quality of service. As expected, objective B 1 achieves the highest operational value and maximizing B 2 objective has the lowest operational value (Figure 8). The operational value achieved by our approach (SM) is closer to that of B 1, with a slightly higher miles driven (Figure 9) and higher user utility. The total number of trips formed by our approach is also comparable to that of B 1. This shows that our approach improves the quality of trips without significantly affecting the total miles driven or the cost of operating the service by the provider. Since matching is performed every 30 seconds, it is important to ensure that the matching algorithm is fast so that it may be effectively used in real-time. The run time (in seconds) of our matching algorithm is 0.5 on average in the zone with high request density (zone 8), 0.12 in zone 28, and 0.003 in zone 56, demonstrating the scalability of DROPS.We also compared our matching algorithm to a hindsight greedy matching with access to all the requests in a day, including future ones. The purpose of this experiment is to evaluate the gain in operational value and user value that could be achieved when knowledge of future requests is available. We compare the total operational value obtained using our approach with that of optimizing only for operational value with all requests in a day. Similarly, the total user value obtained with our approach, with requests arriving in real-time, is compared with that of optimizing for user value only and with access to all requests in a day. Trips are formed using the best-fit greedy algorithm (Algorithm 1) for our approach and for the hindsight evaluation. The , summarized in TAB4, show that our approach achieves at least ∼89% of the operational value and up to ∼84% of the user value compared to the hindsight X X X X X X X X matching in all three zones, indicating that any prediction method of future requests would yield very limited performance gains in the operational value. However, some improvements in user value could be achieved with knowledge of future requests by forming trips where the users have a higher social compatibility with co-passengers. Dynamic ridesharing is an increasingly appealing commuter option. However, numerous surveys have indicated that users' concerns, primarily about the social characteristics of co-passengers, pose a major barrier to using ridesharing for a segment of the population. We present the DROPS system for optimizing dynamic ridesharing with social preferences and present an efficient real-time matching algorithm that can handle effectively high density zones. Our demonstrate that factoring social preferences into the matching process helps improve the user value, without significantly affecting the operational value to the service provider. Furthermore, survey indicate that services that perform social matching are likely to incentivize more individuals to use the service. We conclude that while social matching is beneficial overall, it is not always guaranteed to in improved performance. Factoring social preferences into the matching process is most beneficial in zones with a high request density per decision cycle and greater compatibility among ridesharing users. In the future, we aim to examine ways to extend the matching model to consider nearby trips that have already been dispatched and are currently en-route. We will also consider more complex ways to factor the competing objectives using more general multi-objective planning algorithms BID23. Additionally, based on the performance analysis of our approach with that of a hindsight trip formation, we aim to employ a predictive model for future requests to improve the user value. While we anticipate some performance gains, we do not expect the relative benefits of social matching to diminish.

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We propose a novel dynamic ridesharing framework to form trips that optimizes both operational value for the service provider and user value to the passengers by factoring the users' social preferences into the decision-making process.

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Techniques for combining generalized policies with search algorithms to exploit the strengths and overcome the weaknesses of each when solving probabilistic planning problems