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| MNIST | 0 1 2 | 88.8±6.0 | 91.4±5.1 | 82.3±9.5 | 88.2±6.5 | 82.1±8.5 | 85.9±6.6 | 83.7±8.4 | 86.5±6.8 | 88.8±6.0 90.7±4.4 92.5±3.6 93.9±3.3 | 91.4±5.1 86.0±4.6 87.7±3.8 89.8±3.3 | 89.7±7.5 92.8±5.3 94.9±4.2 | 86.5±6.8 91.2±4.9 92.0±3.6 94.7±2.8 | 83.6±8.2 90.3±4.6 93.9±2.8 |
| F-MNIST | 3 0 1 2 3 | 89.5±6.1 | 89.3±6.2 | 90.2±6.3 | 85.5±7.7 | 81.8±11.2 | 77.8±12.0 | 77.4±11.1 | 76.2±7.3 | 95.5±2.5 89.5±6.1 85.6±7.0 87.8±6.1 89.4±5.5 | 91.9±3.0 89.3±6.2 79.8±9.0 80.1±10.5 83.8±9.4 86.8±7.7 | 96.7±2.3 72.5±15.5 74.3±15.4 77.5±14.7 79.9±13.8 | 97.3±1.8 76.2±7.3 78.2±9.1 80.5±8.2 83.9±7.4 | 96.9±1.7 56.7±4.1 62.3±2.9 67.3±3.0 |
| CIFAR-10 | 5 0 1 2 3 | 60.8±10.7 | 62.9±8.2 | 58.8±10.1 | 59.1±6.6 | 57.3±13.5 | 54.9±11.1 | 55.4±13.3 | 58.6±10.0 | 60.8±10.7 69.8±8.4 73.0±7.1 73.8±6.6 75.1±5.5 | 62.9±8.2 69.3±8.5 72.3±7.5 73.3±7.0 74.2±6.5 | 50.5±3.6 50.3±2.4 50.0±0.7 50.0±1.0 | 58.6±10.0 71.8±7.0 75.2±6.4 77.5±5.9 80.4±4.6 | 60.6±8.3 61.0±6.6 62.7±6.8 60.9±4.6 |
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| MNIST | 0 1 2 | 88.8±6.0 | 91.4±5.1 | 82.3±9.5 | 88.2±6.5 | 82.1±8.5 | 85.9±6.6 | 83.7±8.4 | 86.5±6.8 | 88.8±6.0 90.7±4.4 92.5±3.6 93.9±3.3 | 91.4±5.1 86.0±4.6 87.7±3.8 89.8±3.3 | 89.7±7.5 92.8±5.3 94.9±4.2 | 86.5±6.8 91.2±4.9 92.0±3.6 94.7±2.8 | 83.6±8.2 90.3±4.6 93.9±2.8 |
| F-MNIST | 3 0 1 2 3 | 89.5±6.1 | 89.3±6.2 | 90.2±6.3 | 85.5±7.7 | 81.8±11.2 | 77.8±12.0 | 77.4±11.1 | 76.2±7.3 | 95.5±2.5 89.5±6.1 85.6±7.0 87.8±6.1 89.4±5.5 | 91.9±3.0 89.3±6.2 79.8±9.0 80.1±10.5 83.8±9.4 86.8±7.7 | 96.7±2.3 72.5±15.5 74.3±15.4 77.5±14.7 79.9±13.8 | 97.3±1.8 76.2±7.3 78.2±9.1 80.5±8.2 83.9±7.4 | 96.9±1.7 56.7±4.1 62.3±2.9 67.3±3.0 |
| CIFAR-10 | 5 0 1 2 3 | 60.8±10.7 | 62.9±8.2 | 58.8±10.1 | 59.1±6.6 | 57.3±13.5 | 54.9±11.1 | 55.4±13.3 | 58.6±10.0 | 60.8±10.7 69.8±8.4 73.0±7.1 73.8±6.6 75.1±5.5 | 62.9±8.2 69.3±8.5 72.3±7.5 73.3±7.0 74.2±6.5 | 50.5±3.6 50.3±2.4 50.0±0.7 50.0±1.0 | 58.6±10.0 71.8±7.0 75.2±6.4 77.5±5.9 80.4±4.6 | 60.6±8.3 61.0±6.6 62.7±6.8 60.9±4.6 |
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| F-MNIST | 3 0 1 2 3 | 89.5±6.1 | 89.3±6.2 | 90.2±6.3 | 85.5±7.7 | 81.8±11.2 | 77.8±12.0 | 77.4±11.1 | 76.2±7.3 | 95.5±2.5 89.5±6.1 85.6±7.0 87.8±6.1 89.4±5.5 | 91.9±3.0 89.3±6.2 79.8±9.0 80.1±10.5 83.8±9.4 86.8±7.7 | 96.7±2.3 72.5±15.5 74.3±15.4 77.5±14.7 79.9±13.8 | 97.3±1.8 76.2±7.3 78.2±9.1 80.5±8.2 83.9±7.4 | 96.9±1.7 56.7±4.1 62.3±2.9 67.3±3.0 |
| CIFAR-10 | 5 0 1 2 3 | 60.8±10.7 | 62.9±8.2 | 58.8±10.1 | 59.1±6.6 | 57.3±13.5 | 54.9±11.1 | 55.4±13.3 | 58.6±10.0 | 60.8±10.7 69.8±8.4 73.0±7.1 73.8±6.6 75.1±5.5 | 62.9±8.2 69.3±8.5 72.3±7.5 73.3±7.0 74.2±6.5 | 50.5±3.6 50.3±2.4 50.0±0.7 50.0±1.0 | 58.6±10.0 71.8±7.0 75.2±6.4 77.5±5.9 80.4±4.6 | 60.6±8.3 61.0±6.6 62.7±6.8 60.9±4.6 |
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| Deep belief network, 784-500-500-2000-10 (Hinton, et al.,2012) | | 1.18(0.92 with dropout) |
| Discriminative recurrent sparse auto-encoder1.08(1.21 with 2OO hidden units) 784-40011-10 | | |
| Supervised dictionary learning, 45 × (784-24*) to 45 × (784-96*) (Mairal, et al., 2009) | | 1.05(3.56 without contrastive loss) |
+
+Additional regularizations and boosting-like techniques can further improve performance of networks with LISTA-like encoders. Recent examples include dropout, which trains and then averages over a large set of random subnetworks formed by removing a constant fraction of the hidden units from the original network (Goodfellow, et al., 2013; Hinton, et al., 2012). Deep belief networks and deep Boltzmann machines fine-tuned with dropout are the current state-of-the-art for pixelpermutation-agnostic handwritten digit recognition (Hinton, et al., 2012), and can achieve MNIST classification error as low as $0 . 7 9 \%$ with a carefully tuned network structure and multi-step training procedure. Deep convex networks, which iteratively refine the classification by successively training a stack of classifiers, with the output of the $i - 1 \mathrm { s t }$ classifier provided as input to the ith classifier, can achieve an MNIST error of $0 . 8 3 \%$ (Deng & Yu, 2011). Regularizing by explicit modeling of the data manifold, and then minimizing the square of the Jacobian of the output along the tangent bundle around the training datapoints, can reduce MNIST error to $0 . 8 1 \%$ (Rifai, et al., 2011). Further performance improvements are possible if translation invariance is built directly into the network via a convolutional architecture, and deformations of the inputs are included in the training set (LeCun, et al., 1998), yielding error as low as $0 . 2 3 \%$ (Ciresan, Meier, & Schmidhuber, 2012). These regularizations and augmentations are potentially compatible with DrSAE, but we defer their exploration to future work.
+
+Recurrence is essential to the performance of DrSAEs. If the number of recurrent iterations is decreased from eleven to two, MNIST classification error in a network with 400 hidden units increases from $1 . 0 8 \%$ to $1 . 3 2 \%$ . With only 200 hidden units, MNIST classification error increases from $1 . 2 1 \%$ to $1 . 4 9 \%$ , although the hidden units still differentiate into part-units and categorical-units, as shown in figure 3(d,f).
+
+# 5 Discussion
+
+It is widely believed that natural stimuli, such as images and sounds, fall near a low-dimensional manifold within a higher-dimensional space (the manifold hypothesis) (Bengio, Courville, & Vincent, 2012; Lee, Pedersen, & Mumford, 2003; Olshausen & Field, 2004). The low-dimensional data manifold provides an intuitively compelling and empirically effective basis for classification (Rifai, et al., 2011; Simard, LeCun, & Denker, 1993; Simard, et al., 1998). The continuous deformations that define the data manifold usually preserve identity, whereas even relatively small invalid transformations may change the class of a stimulus. For instance, the various handwritten renditions of the digit 3 in in the last column of figure 9(c) barely overlap, and so the Euclidean distance between them in pixel space is greater than that to the nearest 8 formed by closing both loops. Nevertheless, smooth deformations of one 3 into another correspond to relatively short trajectories along the data manifold,9 whereas the transformation of a 3 into an 8 requires a much longer path within the data manifold. A prohibitive amount of data is required to fully characterize the data manifold (Narayanan & Mitter, 2010), so it is often approximated by the set of linear submanifolds tangent to the data manifold at the observed datapoints, known as the tangent spaces (Ekanadham, Tranchina, & Simoncelli, 2011; Rifai, et al., 2011; Simard, et al., 1998). DrSAEs naturally and efficiently form a tangent space-like representation, consisting of a point on the data manifold indicated by the categorical-units, and a shift within the tangent space specified by the part-units.
+
+Before discriminative fine-tuning, DrSAEs perform a traditional part-based decomposition, familiar from sparse coding, as shown in figure 9(a). The decoding matrix columns are class-independent, local pen strokes, and many units make a comparable, small contribution to the reconstruction. After discriminative fine-tuning, the hidden units differentiate into sparse coding local part-units, and global prototype categorical-units that integrate over them. As shown in figure 9(b,c), the input is decomposed into a prototype, corresponding to a point on the data manifold; and a set of deformations from this prototype along the data manifold, corresponding to shifts within the tangent space. The same prototype can be used for very different inputs, as demonstrated in figure 9(c), since the space of deformations is rich enough to encompass diverse transformations without moving off the data manifold. Even when the prototype is very different from the input, all steps along the reconstruction trajectories in figure 9(b,c) are recognizable as members of the same class.
+
+
+Figure 9: Discriminative recurrent sparse auto-encoders decompose the input into a prototype and deformations along the data manifold. The progressive reconstruction of selected inputs by the hidden representation before (a) or after (b,c) discriminative fine-tuning. The columns from left to right depict either the components of the reconstruction (top row of each pair), or the partial reconstruction induced by the first $n$ parts (bottom row of each pair). Parts are added to the reconstruction in order of decreasing contribution magnitude; smoother transformations are possible with an optimized sequence. The last two columns show the final reconstruction with all parts (Fin), and the original input (Inp). Bars above the decoding matrix columns indicate the scale factor/hidden unit activity associated with the column.
+
+The prototypes learned by the categorical-units for each class are not simply the average over the elements of the class, as depicted in figure 10. Each class includes many possible input variations, so its average is blurry. The prototypes, in contrast, are sharp, and look like representative elements of the appropriate class. Many categorical-units are available for each class, as shown in figure 6. Not all categorical-units correspond to full prototypes; some capture global transformations of a prototype, such as rotations (Simard, et al., 1998).
+
+Consistent with prototypes for the non-negative MNIST inputs, the decoding matrix columns of the categorical-units are generally positive, as shown in figure 7(e). In contrast, the decoders of the part-units are approximately mean-zero and so cannot serve as prototypes themselves. Rather, they shift and transform prototypes, moving activation from one region in the image to another, as demonstrated in figure 9(b,c).
+
+
+Figure 10: The prototypes learned by categorical-units resemble representative instances of the appropriate class, and are sharper than the average over all members of the class in the dataset. The left-most column in each group depicts the average over all elements of each of the ten MNIST digit classes. The other columns show the decoders of the associated units with the largest-magnitude columns in the classification matrix C. Bars above the decoders indicate the angle between the encoder and the decoder for the displayed unit. The most prototypical unit always makes the strongest contribution to the classification, and has a large (but not necessarily the largest) angle between its encoder and decoder. Some units that make large contributions to the classification represent global transformations, such as rotations, of a prototype (Simard, et al., 1998).
+
+Discrepancies between the prototype and the input due to transformations along the data manifold are explained by class-consistent part-units, and only serve to further activate the categorical-units of that class, as in figure 6(a,c). Discrepancies between the prototype and the input due to deformations orthogonal to the data manifold are explained by class-incompatible part-units, and serve to suppress the categorical-units of that class, both directly and via activation of incompatible categorical-units.
+
+If the wrong prototype is turned on, the residual input will generally contain substantial unexplained components. Part-units obey ISTA-like dynamics and thus function as a sparse coder on the residual input, so part-units that match the unexplained components of the input will be activated. These partunits will have positive connections to categorical-units with compatible prototypes, and so will tend to activate categorical-units associated with the true class (so long as the unexplained components of the input are diagnostic). The spuriously activated categorical-unit will not be able to sustain its activity, since few compatible part-units will be required to capture the residual input.
+
+The classification approach used by DrSAEs is different from one based upon a traditional sparse coding decomposition: it projects into the space of deviations from a prototype, which is not the same as the space of prototype-free parts, as is clear from figure 9(a,b). For instance, a 5 can easily be constructed using the parts of a 6, making it difficult to distinguish the two. Indeed, the first seven progressive reconstruction steps of the 6 in figure 9(a) could just as easily be used to produce a 5. However, starting from a 6 prototype, the parts required to break the bottom loop are outside the data manifold of the 6 class, and so will tend to change the active prototype.
+
+DrSAEs naturally learn a hierarchical representation within a recurrent network, thereby implementing a deep network with parameter sharing between the layers.
+
+# References
+
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+Hinton, G. (2010). A practical guide to training restricted Boltzmann machines (UTML TR 2010- 003, version 1). Toronto, Canada: University of Toronto, Department of Computer Science.
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+Lee, A. B., Pedersen, K. S., & Mumford, D. (2003). The nonlinear statistics of high-contrast patches in natural images. International Journal of Computer Vision, 54(1), 83–103.
+Lee, H., Ekanadham, C., & Ng, A. (2008). Sparse deep belief net model for visual area V2. In J. C. Platt, D. Koller, Y. Singer & S. Roweis (Eds.) Advances in Neural Information Processing Systems (NIPS 20), (pp. 873–880).
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+Mairal, J., Bach, F., & Ponce, J. (2012). Task-driven dictionary learning. IEEE Transactions on Pattern Analysis and Machine Intelligence, 34(4), 791–804.
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+Olshausen, B. A., & Field, D. J. (1997). Sparse coding with an overcomplete basis set: A strategy employed by VI? Vision Research, 37(23), 3311–3326.
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+Ranzato M., Poultney, C., Chopra, S., & LeCun, Y. (2006). Efficient learning of sparse representations with an energy-based model. In B. Scholkopf, J. Platt, & T. Hoffman (Eds.) ¨ Advances in Neural Information Processing Systems (NIPS 19), (pp. 1137–1144).
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+Rifai, S., Dauphin, Y., Vincent, P., Bengio, Y., & Muller, X. (2011). The manifold tangent classifier. In J. Shawe-Taylor, R. S. Zemel, P. Bartlett, F. C. N. Pereira, & K. Q. Weinberger (Eds.) Advances in Neural Information Processing Systems (NIPS 24) (pp. 2294–2302).
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+Salinas, E., & Abbott, L. F. (1996). A model of multiplicative neural responses in parietal cortex. Proceedings of the National Academy of Sciences of the United States of America, 93(21), 11956– 11961.
+Seung, H. S. (1998). Learning continuous attractors in recurrent networks. In M. I. Jordan, M. J. Kearns, & S. A. Solla (Eds.) Advances in Neural Information Processing Systems (NIPS 10) (pp. 654–660).
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index 0000000000000000000000000000000000000000..352962238fb849cc91861b427cc498ff292981fd
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+ {
+ "type": "text",
+ "text": "Discriminative Recurrent Sparse Auto-Encoders ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "Jason Tyler Rolfe & Yann LeCun Courant Institute of Mathematical Sciences, New York University 719 Broadway, 12th Floor New York, NY 10003 {rolfe, yann}@cs.nyu.edu ",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "Abstract ",
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+ "bbox": [
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+ "page_idx": 0
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+ {
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+ "text": "We present the discriminative recurrent sparse auto-encoder model, comprising a recurrent encoder of rectified linear units, unrolled for a fixed number of iterations, and connected to two linear decoders that reconstruct the input and predict its supervised classification. Training via backpropagation-through-time initially minimizes an unsupervised sparse reconstruction error; the loss function is then augmented with a discriminative term on the supervised classification. The depth implicit in the temporally-unrolled form allows the system to exhibit far more representational power, while keeping the number of trainable parameters fixed. ",
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+ "text": "From an initially unstructured network the hidden units differentiate into categorical-units, each of which represents an input prototype with a well-defined class; and part-units representing deformations of these prototypes. The learned organization of the recurrent encoder is hierarchical: part-units are driven directly by the input, whereas the activity of categorical-units builds up over time through interactions with the part-units. Even using a small number of hidden units per layer, discriminative recurrent sparse auto-encoders achieve excellent performance on MNIST. ",
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+ "type": "text",
+ "text": "1 Introduction ",
+ "text_level": 1,
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+ "text": "Deep networks complement the hierarchical structure in natural data (Bengio, 2009). By breaking complex calculations into many steps, deep networks can gradually build up complicated decision boundaries or input transformations, facilitate the reuse of common substructure, and explicitly compare alternative interpretations of ambiguous input (Lee, Ekanadham, & $\\mathrm { N g }$ , 2008; Zeiler, Taylor, & Fergus, 2011). Leveraging these strengths, deep networks have facilitated significant advances in solving sensory problems like visual classification and speech recognition (Dahl, et al., 2012; Hinton, Osindero, & Teh, 2006; Hinton, et al., 2012). ",
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+ "type": "text",
+ "text": "Although deep networks have traditionally used independent parameters for each layer, they are equivalent to recurrent networks in which a disjoint set of units is active on each time step. The corresponding representations are sparse, and thus invite the incorporation of powerful techniques from sparse coding (Glorot, Bordes, & Bengio, 2011; Lee, Ekanadham, & Ng, 2008; Olshausen & Field, 1996, 1997; Ranzato, et al., 2006). Recurrence opens the possibility of sharing parameters between successive layers of a deep network. ",
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+ "text": "This paper introduces the Discriminative Recurrent Sparse Auto-Encoder model (DrSAE), comprising a recurrent encoder of rectified linear units (ReLU; Coates & Ng, 2011; Glorot, Bordes, & Bengio, 2011; Jarrett, et al., 2009; Nair & Hinton, 2010; Salinas & Abbott, 1996), connected to two linear decoders that reconstruct the input and predict its supervised classification. The recurrent encoder is unrolled in time for a fixed number of iterations, with the input projecting to each resulting layer, and trained using backpropagation-through-time (Rumelhart, et al., 1986). Training initially minimizes an unsupervised sparse reconstruction error; the loss function is then augmented with a discriminative term on the supervised classification. In its temporally-unrolled form, the network can be seen as a deep network, with parameters shared between the hidden layers. The temporal depth allows the system to exhibit far more representational power, while keeping the number of trainable parameters fixed. ",
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+ "text": "Interestingly, experiments show that DrSAE does not just discover more discriminative “parts” of the form conventionally produced by sparse coding. Rather, the hidden units spontaneously differentiate into two types: a small number of categorical-units and a larger number of part-units. The categorical-units have decoder bases that look like prototypes of the input classes. They are weakly influenced by the input and activate late in the dynamics as the result of interaction with the part-units. In contrast, the part-units are strongly influenced by the input, and encode small transformations through which the prototypes of categorical-units can be reshaped into the current input. Categorical-units compete with each other through mutual inhibition and cooperate with relevant part-units. This can be interpreted as a representation of the data manifold in which the categoricalunits are points on the manifold, and the part-units are akin to tangent vectors along the manifold. ",
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+ "text": "1.1 Prior work ",
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+ "text": "The encoder architecture of DrSAE is modeled after the Iterative Shrinkage and Threshold Algorithm (ISTA), a proximal method for sparse coding (Chambolle, et al., 1998; Daubechies, Defrise, & De Mol, 2004). Gregor & LeCun (2010) showed that the sparse representations computed by ISTA can be efficiently approximated by a structurally similar encoder with a less restrictive, learned parameterization. Rather than learn to approximate a precomputed optimal sparse code, the LISTA autoencoders of Sprechmann, Bronstein, & Sapiro (2012a,b) are trained to directly minimize the sparse reconstruction loss function. DrSAE extends LISTA autoencoders with a non-negativity constraint, which converts the shrink nonlinearity of LISTA into a rectified linear operator; and introduces a unified classification loss, as previously used in conjunction with traditional sparse coders (Bradley & Bagnell, 2008; Mairal, et al., 2009; Mairal, Bach, & Ponce, 2012) and other autoencoders (Boureau, et al., 2010; Ranzato & Szummer, 2008). ",
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+ "text": "DrSAEs resemble the structure of deep sparse rectifier neural networks (Glorot, Bordes, & Bengio, 2011), but differ in that the parameter matrices at each layer are tied (Bengio, BoulangerLewandowski, & Pascanu, 2012), the input projects to all layers, and the outputs are normalized. DrSAEs are also reminiscent of the recurrent neural networks investigated by Bengio & Gingras (1996), but use a different nonlinearity and a heavily regularized loss function. Finally, they are similar to the recurrent networks described by Seung (1998), but have recurrent connections amongst the hidden units, rather than between the hidden units and the input units, and introduce classification and sparsification losses. ",
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+ "text": "In the following, we use lower-case bold letters to denote vectors, upper-case bold letters to denote matrices, superscripts to indicate iterative copies of a vector, and subscripts to index the columns (or rows, if explicitly specified by the context) of a matrix or (without boldface) the elements of a vector. We consider discriminative recurrent sparse auto-encoders (DrSAEs) of rectified linear units with the architecture shown in figure 1: ",
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+ "text": "$$\n\\mathbf { z } ^ { t + 1 } = \\operatorname* { m a x } \\left( 0 , \\mathbf { E } \\cdot \\mathbf { x } + \\mathbf { S } \\cdot \\mathbf { z } ^ { t } - \\mathbf { b } \\right)\n$$",
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+ "text": "for $t = 1 , \\dots , T$ , where $n$ -dimensional vector $\\mathbf { z } ^ { t }$ is the activity of the hidden units at iteration $t$ , $m$ - dimensional vector $\\mathbf { x }$ is the input, and $\\mathbf { z } ^ { t = 0 } = 0$ . Unlike traditional recurrent autoencoders (Bengio, Boulanger-Lewandowski, & Pascanu, 2012), the input projects to every iteration. We call the $n \\times m$ parameter matrix $\\mathbf { E }$ the encoding matrix, and the $n \\times n$ parameter matrix $\\mathbf { S }$ the explaining-away matrix. The $n$ -element parameter vector $\\mathbf { b }$ contains a bias term. The parameters also include the $m \\times n$ decoding matrix $\\mathbf { D }$ and the $l \\times n$ classification matrix $\\mathbf { C }$ . ",
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+ "text": "$$\nL ^ { U } = \\frac { 1 } { 2 } \\cdot \\left| \\left| \\mathbf { x } - \\mathbf { D } \\cdot \\mathbf { z } ^ { T } \\right| \\right| _ { 2 } ^ { 2 } + \\lambda \\cdot \\left| \\left| \\mathbf { z } ^ { T } \\right| \\right| _ { 1 } ,\n$$",
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+ "Figure 1: The discriminative recurrent sparse auto-encoder (DrSAE) architecture. $\\mathbf { z } ^ { t }$ is the hidden representation after iteration $t$ of $T$ , and is initialized to $\\mathbf { z } ^ { 0 } = 0$ ; $\\mathbf { x }$ is the input; and $\\mathbf { y }$ is the supervised classification. Overbars denote approximations produced by the network, rather than the true input. E, S, D, and $\\mathbf { b }$ are learned parameters. "
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+ "text": "with the magnitude of the columns of $\\mathbf { D }$ bounded by 1,1 and the magnitude of the rows of $\\mathbf { E }$ bounded by $\\textstyle { \\frac { 1 . 2 5 } { T } }$ .2 We then add in the supervised classification loss function ",
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+ "type": "equation",
+ "img_path": "images/cab038ed5b4c0920c4fee36c78a038c9eb04ccc45618305bc5892f33665d0194.jpg",
+ "text": "$$\nL ^ { S } = \\mathrm { l o g i s t i c } _ { y } \\left( \\mathbf { C } \\cdot \\frac { \\mathbf { z } ^ { T } } { | | \\mathbf { z } ^ { T } | | } \\right) ,\n$$",
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+ "type": "text",
+ "text": "where the multinomial logistic loss function is defined by ",
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+ "type": "equation",
+ "img_path": "images/e84635ad096825dd38c752437a2f8a4a6418959fd69c70ff9e5190ec9f554539.jpg",
+ "text": "$$\n\\mathrm { l o g i s t i c } _ { y } ( { \\bf z } ) = z _ { y } - \\log \\left( \\sum _ { i } e ^ { z _ { i } } \\right) ,\n$$",
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+ "type": "text",
+ "text": "and $y$ is the index of the desired class.3 Starting with the parameters learned by the unsupervised pretraining, we perform discriminative fine-tune by stochastic gradient descent on $L ^ { U } + \\dot { L } ^ { S }$ , with the magnitude of the rows of $\\mathbf { C }$ bounded by 5.4 The learning rate of each matrix is scaled down by the number of times it is repeated in the network, and the learning rate of the classification matrix is scaled down by a factor of 5, to keep the effective learning rate consistent amongst the parameter matrices. ",
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+ "text": "We train DrSAEs with $T = 1 1$ recurrent iterations (ten nontrivial passes through the explainingaway matrix S)5 and 400 hidden units on the MNIST dataset of $2 8 \\times 2 8$ grayscale handwritten digits (LeCun, et al., 1998), with each input normalized to have $\\ell _ { 2 }$ magnitude equal to 1. We use a training set of 50,000 elements, and a validation set of 10,000 elements to perform early-stopping. Encoding, decoding, and classification matrices learned via this procedure are depicted in figure 2. ",
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+ "type": "text",
+ "text": "The dynamics of equation 1 are inspired by the Learned Iterative Shrinkage and Thresholding Algorithm (LISTA) (Gregor & LeCun, 2010), an efficient approximation to the sparse coding Iterative Shrinkage and Threshold Algorithm (ISTA) (Chambolle, et al., 1998; Daubechies, Defrise, & De ",
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+ "img_path": "images/6b685c9769fce56001f6d528bf3ecefa5b6271c57437781fdad6c1b1ecbf80c4.jpg",
+ "image_caption": [
+ "Figure 2: The hidden units differentiate into spatially localized part-units, which have well-aligned encoders and decoders; and global prototype categorical-units, which have poorly aligned encoders and decoders. A subset of the rows of encoding matrix $\\mathbf { E }$ (a) and the columns of decoding matrix D (b), and all rows of the classification matrix $\\mathbf { C }$ (c) after training. The first row of (a,b) shows the most categorical units; the last row contains the least categorical units; and the middle row evenly steps through the remaining units in order of categoricalness. Gray pixels denote connections with weight 0; darker pixels indicate more positive connections. "
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+ "type": "text",
+ "text": "Mol, 2004). ISTA is an algorithm for minimizing the $\\ell _ { 1 }$ -regularized reconstruction loss function $L ^ { U }$ of equation 2 with respect to ${ \\mathbf z } ^ { T }$ . It is defined by the iterative step ",
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+ "type": "equation",
+ "img_path": "images/1348982e15f03acc1e429ff9dbee174a5ae5f37663fd736a838e980808f4b4b7.jpg",
+ "text": "$$\n{ \\bf z } ^ { t + 1 } = h _ { \\alpha \\cdot \\lambda } \\left( \\alpha \\cdot { \\bf D } ^ { \\top } \\cdot { \\bf x } + \\left( { \\bf I } - \\alpha \\cdot { \\bf D } ^ { \\top } \\cdot { \\bf D } \\right) \\cdot { \\bf z } ^ { t } \\right) ,\n$$",
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+ "type": "text",
+ "text": "where $\\left[ h _ { \\theta } ( \\mathbf { x } ) \\right] _ { i } = { \\mathrm { s i g n } } \\left( x _ { i } \\right) \\cdot { \\mathrm { m a x } } \\left( 0 , | x _ { i } | - \\theta \\right)$ and $\\alpha$ is a small step-size parameter. With nonnegative units, ISTA is equivalent to projected gradient descent of $L ^ { U }$ of equation 2. As the number of iterations $T \\to \\infty$ , a DrSAE defined by equation 1 becomes a non-negative version of ISTA if it satisfies the restrictions: ",
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+ {
+ "type": "equation",
+ "img_path": "images/354692d1341d530e9d1b0d4478df9543fbc819d37cfd8a942a7359523a1a7a12.jpg",
+ "text": "$$\n{ \\bf E } = \\alpha \\cdot { \\bf D } ^ { \\top } , \\qquad { \\bf S } = { \\bf I } - \\alpha \\cdot { \\bf D } ^ { \\top } \\cdot { \\bf D } , \\qquad b _ { i } = \\alpha \\cdot \\lambda , \\quad \\mathrm { ~ a n d ~ } \\quad z _ { i } ^ { t } \\geq 0 ,\n$$",
+ "text_format": "latex",
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+ "type": "text",
+ "text": "where the positive scale factor $\\alpha$ is less than the maximal eigenvalue of $\\mathbf { D } ^ { \\top } \\cdot \\mathbf { D }$ , and $\\mathbf { I }$ is the $n \\times n$ identity matrix. ",
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+ "type": "text",
+ "text": "As in LISTA, but unlike ISTA, the encoding matrix $\\mathbf { E }$ and explaining-away matrix S in a DrSAE are independent of the decoding matrix $\\mathbf { D }$ . Connections from the input to the hidden units, and recurrent connections between the hidden units, are all-to-all, so the network structure is agnostic to permutations of the input. DrSAEs can also be understood as deep, feedforward networks with the parameter matrices tied between the layers. ",
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+ {
+ "type": "text",
+ "text": "3 Analysis of the hidden unit representation ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "Discriminative fine-tuning naturally induces the hidden units of a DrSAE to differentiate into a hierarchy-like continuum. On one extreme are part-units, which perform an ISTA-like sparse coding computation; on the other are categorical-units, which use a sophisticated form of pooling to integrate over matching part-units, and implement winner-take-all dynamics amongst themselves. Converging lines of evidence indicate that these two groups use distinct computational mechanisms and serve different representational roles. ",
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+ "type": "text",
+ "text": "In the ISTA algorithm, each row of the encoding matrix $\\mathbf { E } _ { i }$ (which we sometimes call the encoder of unit $i$ ) is proportional to the corresponding column of the decoding matrix $\\mathbf { D } _ { i }$ (which we call the decoder of unit $i$ ), and each row $( \\mathbf { S } - \\mathbf { I } ) _ { i }$ is proportional to $\\left( \\mathbf { D } _ { i } \\right) ^ { \\top } \\cdot \\bar { \\mathbf { D } }$ , as in equation 4. As a result, the angle between $\\mathbf { E } _ { i }$ and $\\mathbf { D } _ { i }$ , and the angle between the rows of $\\mathbf { S } - \\mathbf { I }$ and $\\mathbf { D } ^ { \\top } \\cdot \\mathbf { D }$ , are both simple measures of the degree to which a unit’s dynamics follow the ISTA algorithm, and thus perform sparse coding.6 These quantities are equal to 0 in the case of perfect ISTA, and grow larger as the network diverges from ISTA. Of these two angles, the explaining-away matrix comparison is more difficult to interpret, since a distortion of any one unit’s decoding column $\\mathbf { D } _ { i }$ will affect all rows of $\\mathbf { D } ^ { \\top } \\cdot \\mathbf { D }$ , whereas the angle between the encoder row $\\mathbf { E } _ { i }$ and decoder column $\\mathbf { D } _ { i }$ only depends upon a single unit. For this reason, we use the angle between the encoder row and decoder column as a measure of the position of each unit on the part/categorical continuum. ",
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+ "img_path": "images/86af5b007186604e48419daa5cda075306d95ffc27e1b7c5feb6c357aa149449.jpg",
+ "image_caption": [
+ "Figure 3: The hidden units differentiate into two populations after discriminative fine-tuning. The magnitude of row $( \\mathbf { S } - \\mathbf { I } ) _ { i }$ (a,b,e) and $\\mathbf { C } _ { i }$ (c,d), versus the angle between encoder row and decoder column, for each unit from networks using 11 (a,c,e) and 2 (b,d,f) iterations. All plots are from discriminatively fine-tuned networks except (a,b), which are only subject to unsupervised pretraining. We call the dense cloud in the bottom-left part-units, and the tail extending to the top-right categorical-units. "
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+ "text": "",
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+ "type": "text",
+ "text": "Figure 3 plots, for each unit $i$ , the magnitude of row $( \\mathbf { S } - \\mathbf { I } ) _ { i }$ and column $\\mathbf { C } _ { i }$ , versus the angle between row $\\mathbf { E } _ { i }$ and column $\\mathbf { D } _ { i }$ . Before discriminative fine-tuning, there are no categorical-units; the angle between the encoder row and decoder column is small and the incoming recurrent connections are weak for all units, as in figure 3(a,b). After discriminative fine-tuning, there remains a dense cloud of points for which the angle between the encoder row and decoder column is very small, and the incoming recurrent and outgoing classification connections are weak. Abutting this is an extended tail of points that have a larger angle between the encoder row and decoder column, and stronger incoming recurrent and outgoing classification connections. We call units composing the dense cloud part-units, since they have ISTA-compatible connections, while we refer to those making up the extended tail as categorical-units, since they have strong connections to the classification output.7 When trained on MNIST, part-units have localized, pen stroke-like decoders, as can be seen in the bottom rows of figure 2(a,b). Categorical-units, in contrast, tend to have whole-digit prototype-like decoders, as in the top rows of figure 2(a,b). Discriminative fine-tuning induces the differentiation of categorical-units regardless of the depth of the encoder. ",
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+ {
+ "type": "text",
+ "text": "3.1 Part-units ",
+ "text_level": 1,
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+ {
+ "type": "text",
+ "text": "Examination of the relationship between the elements of $\\mathbf { S } - \\mathbf { I }$ and $\\mathbf { D } ^ { \\top } \\cdot \\mathbf { D }$ confirms that partunits with an encoder-decoder angle less than 0.5 radians abide by ISTA, and so perform sparse coding on the residual input after the categorical-unit prototypes are subtracted out. The prominent diagonals with matching slopes in figure 4(a,b), which plot the value of $S _ { i , j } - \\delta _ { i , j }$ versus $\\mathbf { D } _ { i } \\cdot \\mathbf { D } _ { j }$ for connections between part-units, and from categorical-units to part-units, respectively, demonstrate that part-units receive ISTA-consistent connections from all units. The fidelity of these connections to the ISTA ideal is not strongly dependent upon whether the afferent units are ISTA-compliant partunits, or ISTA-ignoring categorical-units. As a result, the part-units treat the categorical-units as if they were also participating in the reconstruction of the input, and only attempt to reconstruct the residual input not explained by the categorical-unit prototypes. ",
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+ "type": "text",
+ "text": "As can be seen in figure 4(c), the degree to which the encoder conforms to the ISTA algorithm is strongly correlated with the degree to which the explaining-away matrix matches the ISTA algorithm. Figure 5 shows the decoders associated with the strongest recurrent connections to three representative part-units. As expected, the decoders of these afferent units tend to be strongly aligned or anti-aligned with their target’s decoder, and include both part-units and categorical-units. ",
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+ "type": "text",
+ "text": "3.2 Categorical-units ",
+ "text_level": 1,
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+ "type": "text",
+ "text": "In contrast, the recurrent connections to categorical-units with an encoder-decoder angle greater than 0.7 radians are not strongly correlated with the values predicted by ISTA. Rather than analyzing connections to the categorical-units only based upon their destination, it is more informative to consider them organized by their source. Part-units are compatible with categorical-units of certain classes,8 and not with others, as shown by figure 6(a). Part-units generally have positive connections to categorical-units with parallel prototypes, independent of offset, and negative connections to categorical-units with orthogonal prototypes, as shown in figure 7(a). This corresponds to a sophisticated form of pooling (Jarrett, et al., 2009), with a single categorical-unit drawing excitation from a large collection of parallel but not necessarily perfectly aligned part-units, as in figure 6(c). It is also suggestive of the standard Hubel and Wiesel model of complex cells in primary visual cortex (Hubel & Wiesel, 1962). ISTA would instead predict a connection proportional to the inner product, which is zero for orthogonal prototypes and negative for anti-aligned prototypes. ",
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+ "text": "Part-units use sparse coding dynamics, and so are not disproportionately suppressed by categoricalunits that represent any particular class. However, each part-unit is itself compatible with (i.e., has positive connections to) categorical-units of only a subset of the classes. As a result, the categorical",
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+ "img_path": "images/8a7eda654d6effc337db33920084369f97675c71adbfe73004cec8d0e6a94241.jpg",
+ "image_caption": [
+ "Figure 4: Part-units have connections consistent with ISTA. The actual connection weights $\\mathbf { S } - \\mathbf { I }$ versus the ISTA-predicted weights $\\mathbf { D } ^ { \\top } \\cdot \\mathbf { D }$ , for connections from part-units to part-units (a) and categorical-units to part-units (b); and the angle between the rows of $\\mathbf { S } - \\mathbf { I }$ and the ISTA-ideal $\\mathbf { D } ^ { \\top } \\cdot \\mathbf { D }$ versus the angle between the encoder rows and decoder columns (c). Units are considered part-units if the angle between their encoder and decoder is less than 0.5 radians, and categoricalunits if the angle between their encoder and decoder is greater than 0.7 radians. "
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+ "text": "Dest Source units ",
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+ "image_caption": [
+ "Figure 5: Part-units receive ISTA-compatible connections and thus perform sparse coding on the residual input after the contribution of the categorical-units is subtracted out. The decoders of the twenty units with the strongest explaining-away connections $\\lvert S _ { i , j } - \\delta _ { i , j } \\rvert$ to three typical part-units, sorted by connection magnitude. The left-most column depicts the decoder of the recipient partunit. The bars above the decoders in the remaining columns indicate the strength of the connections. Black bars are used for positive connections, and white bars for negative connections. "
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+ "type": "text",
+ "text": "units and thus the class chosen are determined by the part-unit activations. In particular, only a subset of the possible deformations implemented by part-unit decoders are freely available for each prototype, since part-units with a strong negative connection to a categorical-unit will tend to silence it, and so cannot be used to transform the prototype of that categorical-unit. ",
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+ "text": "Categorical-units implement winner-take-all-like dynamics amongst themselves, as shown in figure 6(b), with negative connections to most other categorical-units. Positive total self-connections $S _ { i , i }$ facilitate the integration of inputs over time. ",
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+ {
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+ "img_path": "images/00264b46e1d05b476812510d277021bfc8d82e2cd578be8bbca94e825c6b6ab1.jpg",
+ "image_caption": [
+ "Figure 6: Categorical-units execute a sophisticated form of pooling over part-units, and have winnertake-all dynamics amongst themselves. The decoders of the categorical-units receiving the twenty strongest connections $| \\bar { S } _ { i , j } - \\delta _ { i , j } |$ from representative part-units (a) and categorical-units (b), and the decoders of the part-units sending the twenty strongest projections to representative categoricalunits (c). The connections are sorted first by the class of their destination, and then by the magnitude of the connection. The left-most column depicts the decoder of the source (a,b) or destination (c) unit. The bars above the decoders in the remaining columns indicate the strength of the connections. Black bars are used for positive connections, and white bars for negative connections. "
+ ],
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+ "type": "text",
+ "text": "When activated, the categorical-units make a much larger contribution to the reconstruction than any single part-unit, as can be seen in figure 7(b). Since, the projections from categorical-units to part-units are consistent with ISTA, the magnitude of the categorical-unit contribution to the reconstruction need not be tightly regulated. The part-units adjust accordingly to accommodate whatever residual is left by the categorical-units. ",
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+ "type": "text",
+ "text": "The units form a rough hierarchy, with part-units on the bottom and categorical-units on the top. Categorical-units receive strong recurrent connections, as shown in figure 3(c,d) implying that their activity is more determined by other hidden units and less by the input (since the magnitude of the input connections is bounded), and thus they are higher in the hierarchy. As shown in figure 7(c), part-units receive most of their input from other part-units; categorical-units receive a larger fraction of their input from other categorical-units. Whereas part-units have well-structured encoders and are generally activated directly by the input on the first iteration, categorical-units are more likely to first achieve a non-zero activation on the second iteration, as shown in figure 7(d), suggesting that they require stimulation from part-units. The immediate response of part-units in contrast to the gradual refinement of categorical-units is apparent in figure 8, which shows the optimal decoding matrix for selected units, inferred from their observed activity at each iteration. ",
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+ "type": "text",
+ "text": "4 Performance ",
+ "text_level": 1,
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+ "text": "The comparison of MNIST classification performance in table 1 demonstrates the power of the hierarchical representation learned by DrSAEs. Rather than learn to minimize the sum of equations 2 and 3, Gregor & LeCun (2010) train the LISTA encoder to approximate the code generated by a traditional sparse coder. While they do not report classification performance using LISTA, Gregor and LeCun do evaluate MNIST classification error using the related learned coordinate descent algorithm. Sprechmann, Bronstein, & Sapiro (2012a,b) extend this approach by training a LISTA auto-encoder to reconstruct the input directly, using loss functions similar to equation 2. Although they identify the possibility of using regularization dependent upon supervised information, Sprechmann and colleagues do not consider a parameterized classifier operating on a common hidden representation. Instead, they train a separate encoder for each class, and classify each input based upon the encoder with the lowest sparse coding error. DrSAEs significantly outperform these other techniques based upon a LISTA encoder. ",
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+ "image_caption": [
+ "Figure 7: Statistics of connections indicate the presence of a rough hierarchy, with categorical-units on the top integrating over part-units on the bottom. Average explaining-away connection weight $S _ { i j }$ , binned by alignment between decoders, for connections from part-units to categorical-units (a). If no units fall in a given bin, the average is set to zero. Average final value of a unit $z _ { i } ^ { t = T }$ , given that $z _ { i } ^ { t = T } > 0$ , versus the angle between the encoder row $E _ { i }$ and decoder column $D _ { i }$ (b). Average angle between encoder row $E _ { j }$ and decoder column $D _ { j }$ of afferents to unit $i$ , weighted by the strength of the connection to unit $i$ , versus the angle between encoder row $E _ { i }$ and decoder column $D _ { i }$ (c). Probability that $z _ { i } ^ { 1 } = 0$ and $z _ { i } ^ { 2 } > 0$ , versus the angle between the encoder row $E _ { i }$ and decoder column $D _ { i }$ (d). Average value of the decoder column $\\overline { { D _ { i } } }$ versus the angle between the encoder row $E _ { i }$ and the decoder column $D _ { i }$ (e). "
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+ "text": "DrSAEs also perform well compared to other techniques using encoders related to LISTA. Deep sparse rectifier neural networks (Glorot, Bordes, & Bengio, 2011) combine discriminative training with an encoder similar to LISTA, but do not tie the parameters between the layers and only allow the input to project to the first layer. Differentiable sparse coding (Bradley & Bagnell, 2008) and supervised dictionary learning (Mairal, et al., 2009) also train discriminatively, but effectively use an infinite-depth ISTA-like encoder, and are thus much less computationally efficient than DrSAEs. Supervised dictionary learning achieves performance statistically indistinguishable from DrSAEs using a contrastive loss function. A similar technique achieves MNIST classification error as low as $0 . 5 4 \\%$ when the dataset is augmented with shifted copies of the inputs (Mairal, Bach, & Ponce, 2012). ",
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+ "Figure 8: Part-units (a) respond to the input quickly, while the activity of categorical-units (b) refines slowly. Columns of the optimal decoding matrices $\\mathbf { D } ^ { t }$ minimizing the input reconstruction error $| | \\mathbf { x } - \\mathbf { D } ^ { t } \\cdot \\mathbf { z } ^ { t } | | _ { 2 } ^ { 2 }$ from the hidden representation $\\mathbf { z } ^ { t }$ for $t = 1 , \\dots , T$ . The first and last columns show the corresponding encoder and decoder for the chosen representative units. Intermediate columns represent successive iterations $t$ . "
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+ "Table 1: MNIST classification error rate $( \\% )$ for pixel-permutation-agnostic encoders without boosting-like augmentations. The first column indicates the size of each layer in the specified encoder, separated by hyphens. Exponents specify the number of recurrent iterations; asterisks denote repetition to convergence. $1 0 \\times ( \\cdots )$ indicates that a separate encoder is trained for each input class; $4 5 \\times ( \\cdots )$ indicates that a separate encoder is trained for each pairwise binary classification problem. Further performance improvements have been reported with regularization techniques such as dropout, architectures that enforce translation-invariance, and datasets augmented by deformations, as discussed in the main text. "
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+ "table_body": "| LISTA auto-encoder,10 × (289-1005) (Sprechmann,Bronstein,& Sapiro,2012a) | | 3.76(5.98 with 289 hidden units) |
| Learned coordinate descent, 784-78450-10 (Gregor & LeCun, 2010) | 2.29 | |
| Differentiable sparse coding,180-256*-10 (Bradley & Bagnell, 2008) | 1.30 | 1.20(1.16 with tanh nonlinearity) |
| Deep sparse rectifier neural network 784-1000-1000-1000-10 (Glorot, Bordes,& Bengio,2011) | | |
| Deep belief network, 784-500-500-2000-10 (Hinton, et al.,2012) | | 1.18(0.92 with dropout) |
| Discriminative recurrent sparse auto-encoder1.08(1.21 with 2OO hidden units) 784-40011-10 | | |
| Supervised dictionary learning, 45 × (784-24*) to 45 × (784-96*) (Mairal, et al., 2009) | | 1.05(3.56 without contrastive loss) |
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+ "text": "Additional regularizations and boosting-like techniques can further improve performance of networks with LISTA-like encoders. Recent examples include dropout, which trains and then averages over a large set of random subnetworks formed by removing a constant fraction of the hidden units from the original network (Goodfellow, et al., 2013; Hinton, et al., 2012). Deep belief networks and deep Boltzmann machines fine-tuned with dropout are the current state-of-the-art for pixelpermutation-agnostic handwritten digit recognition (Hinton, et al., 2012), and can achieve MNIST classification error as low as $0 . 7 9 \\%$ with a carefully tuned network structure and multi-step training procedure. Deep convex networks, which iteratively refine the classification by successively training a stack of classifiers, with the output of the $i - 1 \\mathrm { s t }$ classifier provided as input to the ith classifier, can achieve an MNIST error of $0 . 8 3 \\%$ (Deng & Yu, 2011). Regularizing by explicit modeling of the data manifold, and then minimizing the square of the Jacobian of the output along the tangent bundle around the training datapoints, can reduce MNIST error to $0 . 8 1 \\%$ (Rifai, et al., 2011). Further performance improvements are possible if translation invariance is built directly into the network via a convolutional architecture, and deformations of the inputs are included in the training set (LeCun, et al., 1998), yielding error as low as $0 . 2 3 \\%$ (Ciresan, Meier, & Schmidhuber, 2012). These regularizations and augmentations are potentially compatible with DrSAE, but we defer their exploration to future work. ",
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+ "text": "Recurrence is essential to the performance of DrSAEs. If the number of recurrent iterations is decreased from eleven to two, MNIST classification error in a network with 400 hidden units increases from $1 . 0 8 \\%$ to $1 . 3 2 \\%$ . With only 200 hidden units, MNIST classification error increases from $1 . 2 1 \\%$ to $1 . 4 9 \\%$ , although the hidden units still differentiate into part-units and categorical-units, as shown in figure 3(d,f). ",
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+ "text": "5 Discussion ",
+ "text_level": 1,
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+ "text": "It is widely believed that natural stimuli, such as images and sounds, fall near a low-dimensional manifold within a higher-dimensional space (the manifold hypothesis) (Bengio, Courville, & Vincent, 2012; Lee, Pedersen, & Mumford, 2003; Olshausen & Field, 2004). The low-dimensional data manifold provides an intuitively compelling and empirically effective basis for classification (Rifai, et al., 2011; Simard, LeCun, & Denker, 1993; Simard, et al., 1998). The continuous deformations that define the data manifold usually preserve identity, whereas even relatively small invalid transformations may change the class of a stimulus. For instance, the various handwritten renditions of the digit 3 in in the last column of figure 9(c) barely overlap, and so the Euclidean distance between them in pixel space is greater than that to the nearest 8 formed by closing both loops. Nevertheless, smooth deformations of one 3 into another correspond to relatively short trajectories along the data manifold,9 whereas the transformation of a 3 into an 8 requires a much longer path within the data manifold. A prohibitive amount of data is required to fully characterize the data manifold (Narayanan & Mitter, 2010), so it is often approximated by the set of linear submanifolds tangent to the data manifold at the observed datapoints, known as the tangent spaces (Ekanadham, Tranchina, & Simoncelli, 2011; Rifai, et al., 2011; Simard, et al., 1998). DrSAEs naturally and efficiently form a tangent space-like representation, consisting of a point on the data manifold indicated by the categorical-units, and a shift within the tangent space specified by the part-units. ",
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+ "text": "Before discriminative fine-tuning, DrSAEs perform a traditional part-based decomposition, familiar from sparse coding, as shown in figure 9(a). The decoding matrix columns are class-independent, local pen strokes, and many units make a comparable, small contribution to the reconstruction. After discriminative fine-tuning, the hidden units differentiate into sparse coding local part-units, and global prototype categorical-units that integrate over them. As shown in figure 9(b,c), the input is decomposed into a prototype, corresponding to a point on the data manifold; and a set of deformations from this prototype along the data manifold, corresponding to shifts within the tangent space. The same prototype can be used for very different inputs, as demonstrated in figure 9(c), since the space of deformations is rich enough to encompass diverse transformations without moving off the data manifold. Even when the prototype is very different from the input, all steps along the reconstruction trajectories in figure 9(b,c) are recognizable as members of the same class. ",
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+ "Figure 9: Discriminative recurrent sparse auto-encoders decompose the input into a prototype and deformations along the data manifold. The progressive reconstruction of selected inputs by the hidden representation before (a) or after (b,c) discriminative fine-tuning. The columns from left to right depict either the components of the reconstruction (top row of each pair), or the partial reconstruction induced by the first $n$ parts (bottom row of each pair). Parts are added to the reconstruction in order of decreasing contribution magnitude; smoother transformations are possible with an optimized sequence. The last two columns show the final reconstruction with all parts (Fin), and the original input (Inp). Bars above the decoding matrix columns indicate the scale factor/hidden unit activity associated with the column. "
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+ "text": "The prototypes learned by the categorical-units for each class are not simply the average over the elements of the class, as depicted in figure 10. Each class includes many possible input variations, so its average is blurry. The prototypes, in contrast, are sharp, and look like representative elements of the appropriate class. Many categorical-units are available for each class, as shown in figure 6. Not all categorical-units correspond to full prototypes; some capture global transformations of a prototype, such as rotations (Simard, et al., 1998). ",
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+ "text": "Consistent with prototypes for the non-negative MNIST inputs, the decoding matrix columns of the categorical-units are generally positive, as shown in figure 7(e). In contrast, the decoders of the part-units are approximately mean-zero and so cannot serve as prototypes themselves. Rather, they shift and transform prototypes, moving activation from one region in the image to another, as demonstrated in figure 9(b,c). ",
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+ "Figure 10: The prototypes learned by categorical-units resemble representative instances of the appropriate class, and are sharper than the average over all members of the class in the dataset. The left-most column in each group depicts the average over all elements of each of the ten MNIST digit classes. The other columns show the decoders of the associated units with the largest-magnitude columns in the classification matrix C. Bars above the decoders indicate the angle between the encoder and the decoder for the displayed unit. The most prototypical unit always makes the strongest contribution to the classification, and has a large (but not necessarily the largest) angle between its encoder and decoder. Some units that make large contributions to the classification represent global transformations, such as rotations, of a prototype (Simard, et al., 1998). "
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+ "text": "Discrepancies between the prototype and the input due to transformations along the data manifold are explained by class-consistent part-units, and only serve to further activate the categorical-units of that class, as in figure 6(a,c). Discrepancies between the prototype and the input due to deformations orthogonal to the data manifold are explained by class-incompatible part-units, and serve to suppress the categorical-units of that class, both directly and via activation of incompatible categorical-units. ",
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+ "text": "If the wrong prototype is turned on, the residual input will generally contain substantial unexplained components. Part-units obey ISTA-like dynamics and thus function as a sparse coder on the residual input, so part-units that match the unexplained components of the input will be activated. These partunits will have positive connections to categorical-units with compatible prototypes, and so will tend to activate categorical-units associated with the true class (so long as the unexplained components of the input are diagnostic). The spuriously activated categorical-unit will not be able to sustain its activity, since few compatible part-units will be required to capture the residual input. ",
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+ "text": "The classification approach used by DrSAEs is different from one based upon a traditional sparse coding decomposition: it projects into the space of deviations from a prototype, which is not the same as the space of prototype-free parts, as is clear from figure 9(a,b). For instance, a 5 can easily be constructed using the parts of a 6, making it difficult to distinguish the two. Indeed, the first seven progressive reconstruction steps of the 6 in figure 9(a) could just as easily be used to produce a 5. However, starting from a 6 prototype, the parts required to break the bottom loop are outside the data manifold of the 6 class, and so will tend to change the active prototype. ",
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+ "text": "DrSAEs naturally learn a hierarchical representation within a recurrent network, thereby implementing a deep network with parameter sharing between the layers. ",
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+ "type": "text",
+ "text": "References ",
+ "text_level": 1,
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+ "page_idx": 12
+ },
+ {
+ "type": "text",
+ "text": "Bengio, Y. (2009). Learning deep architectures for AI. Foundations and Trends in Machine Learning, 2(1), 1–127. \nBengio, Y., Boulanger-Lewandowski, N., & Pascanu, R. (2012). Advances in optimizing recurrent networks. arXiv:1212.0901v2 [cs.LG] \nBengio, Y., Courville, A., & Vincent, P. (2012). Representation learning: A review and new perspectives. arXiv:1206.5538 [cs.LG] \nBengio, Y., & Gingras, F. (1996). Recurrent neural networks for missing or asynchronous data. In D. Touretzky, M. Mozer, & M. Hasselmo (Eds.) Advances in Neural Information Processing Systems (NIPS 8) (pp. 395–401). \nBradley, D. M., & Bagnell, J. A. (2008). Differentiable sparse coding. In D. Koller, D. Schuurmans, Y. Bengio, & L. Bottou (Eds.) Advances in Neural Information Processing Systems (NIPS 21) (pp. 113–120). \nBoureau, Y., Bach, F., LeCun, L., & Ponce, J. (2010). Learning mid-level features for recognition In Proceedings of the 23rd IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2010). \nChambolle, A., De Vore, R. A., Lee, N. Y., & Lucier, B. J. (1998). Nonlinear wavelet image processing: Variational problems, compression, and noise removal through wavelet shrinkage. IEEE Transactions on Image Processing, 7(3), 319–335. \nCiresan, D., Meier, U., & Schmidhuber, J. (2012). Multi-column deep neural networks for image classification. In Proceedings of the 25th IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2012) (pp. 3642–3649). \nCoates, A., & Ng, A. Y. (2011). The importance of encoding versus training with sparse coding and vector quantization L. Getoor & T. Scheffer (Eds.) Proceedings of the 28th International Conference on Machine Learning (ICML 2011) (pp. 921–928). \nDahl, G. E., Yu, D., Deng, L., & Acero, A. (2012). Context-dependent pre-trained deep neural networks for large-vocabulary speech recognition. IEEE Transactions on Audio, Speech, and Language Processing, 20(1), 30–42. \nDaubechies, I., Defrise, M., & De Mol, C. (2004). An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Communications on Pure and Applied Mathematics, 57(11), 1413–1457. \nEkanadham, C., Tranchina, D., & Simoncelli, E. P. (2011). Recovery of sparse translation-invariant signals with continuous basis pursuit. IEEE Transactions on Signal Processing, 59(10), 4735– 4744. \nGlorot, X., Bordes, A., & Bengio, Y. (2011). Deep sparse rectifier neural networks. In G. Gordon, D. Dunson, & M. Dudik (Eds.) JMLR W&CP: Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics (AISTATS 2011) (pp. 315–323). \nGoodfellow, I. J., Warde-Farley, D., Mirza, M., Courville, A., & Bengio, Y. (2013). Maxout networks arXiv:1302.4389v3 [stat.ML] \nGregor, K., & LeCun, Y. (2010). Learning fast approximations of sparse coding. In J. Furnkranz ¨ & T. Joachims (Eds.) Proceedings of the 27th International Conference on Machine Learning (ICML 2010) (pp. 399–406). \nHinton, G. E., Osindero, S., & Teh, Y. W. (2006). A fast learning algorithm for deep belief nets. Neural Computation, 18(7), 1527–1554. \nHinton, G. (2010). A practical guide to training restricted Boltzmann machines (UTML TR 2010- 003, version 1). Toronto, Canada: University of Toronto, Department of Computer Science. \nHinton, G. E., Srivastava, N., Krizhevsky, A., Sutskever, I., & Salakhutdinov, R. R. (2012). Improving neural networks by preventing co-adaptation of feature detectors arXiv:1207.0580v1 [cs.NE] \nHubel, D. H., & Wiesel, T. N. (1962). Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex. 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Bottou (Eds.) Advances in Neural Information Processing Systems (NIPS 21) (pp. 1033–1040). \nMairal, J., Bach, F., & Ponce, J. (2012). Task-driven dictionary learning. IEEE Transactions on Pattern Analysis and Machine Intelligence, 34(4), 791–804. \nNair, V., & Hinton, G. E. (2010). Rectified linear units improve restricted boltzmann machines. In J. Furnkranz & T. Joachims (Eds.) ¨ Proceedings of the 27th International Conference on Machine Learning (ICML 2010) (pp. 807-814). \nNarayanan, H. & MItter, S. (2010). Sample complexity of testing the manifold hypothesis. In J. Lafferty, C. K. I. Williams, J. Shawe-Taylor, R.S. Zemel, & A. Culotta (Eds.) Advances in Neural Information Processing Systems (NIPS 23) (pp. 1786–1794). \nOlshausen, B. A., & Field, D. J. (1996). Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature, 381(6583), 607–609. \nOlshausen, B. A., & Field, D. J. (1997). 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Advances in Neural Information Processing Systems (NIPS 24) (pp. 2294–2302). \nRumelhart, D. E., Hinton, G. E., & Williams, R. J. (1986). Learning internal representations by error propagation. In D. E. Rumelhart, J. L. McClelland, and the PDP Research Group (Eds.), Parallel Distributed Processing: Explorations in the Microstructure of Cognition: Vol. 1. Foundations (pp. 318–362). Cambridge, MA: MIT Press. \nSalinas, E., & Abbott, L. F. (1996). A model of multiplicative neural responses in parietal cortex. Proceedings of the National Academy of Sciences of the United States of America, 93(21), 11956– 11961. \nSeung, H. S. (1998). Learning continuous attractors in recurrent networks. In M. I. Jordan, M. J. Kearns, & S. A. Solla (Eds.) Advances in Neural Information Processing Systems (NIPS 10) (pp. 654–660). \nSimard, P., LeCun, Y., & Denker, J. S. (1993). Efficient pattern recognition using a new transformation distance. In S. J. Hanson, J. D. Cowan, & C. L. Giles (Eds.) Advances in Neural Information Processing Systems (NIPS 5) (pp. 50–58). \nSimard, P., LeCun, Y., Denker, J., & Victorri, B. (1998). Transformation invariance in pattern recognition: Tangent distance and tangent propagation. In G. Orr, & K. Muller (Eds.), Neural networks: Tricks of the trade. Berlin: Springer. \nSprechmann, P., Bronstein, A., & Sapiro, G. (2012). Learning efficient structured sparse models. In J. Langford & J. Pineau (Eds.) Proceedings of the 29th International Conference on Machine Learning (ICML 12) (pp. 615–622). \nSprechmann, P., Bronstein, A., & Sapiro, G. (2012). Learning efficient sparse and low rank models. arXiv:1212.3631 [cs.LG] \nDeng, L. & Yu, D. (2011). Deep convex net: A scalable architecture for speech pattern classification. In Proceedings of the 12th Annual Conference of the International Speech Communication Association (INTERSPEECH 2011) (pp. 2285-2288). \nZeiler, M. D., Taylor, G. W., & Fergus, R. (2011). Adaptive deconvolutional networks for mid and high level feature learning. In Proceedings of the 13th International Conference on Computer Vision (ICCV 2011) (pp. 2018–2025). ",
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+# DIVIDE-AND-CONQUER REINFORCEMENT LEARNING
+
+Dibya Ghosh1, Avi Singh1, Aravind Rajeswaran2, Vikash Kumar2, Sergey Levine1
+
+1 University of California Berkeley 2 University of Washington Seattle dibya@berkeley.edu, avisingh@cs.berkeley.edu, {aravraj, vikash}@cs.washington.edu, svlevine@eecs.berkeley.edu
+
+# ABSTRACT
+
+Standard model-free deep reinforcement learning (RL) algorithms sample a new initial state for each trial, allowing them to optimize policies that can perform well even in highly stochastic environments. However, problems that exhibit considerable initial state variation typically produce high-variance gradient estimates for model-free RL, making direct policy or value function optimization challenging. In this paper, we develop a novel algorithm that instead partitions the initial state space into “slices”, and optimizes an ensemble of policies, each on a different slice. The ensemble is gradually unified into a single policy that can succeed on the whole state space. This approach, which we term divide-and-conquer $R L$ , is able to solve complex tasks where conventional deep RL methods are ineffective. Our results show that divide-and-conquer RL greatly outperforms conventional policy gradient methods on challenging grasping, manipulation, and locomotion tasks, and exceeds the performance of a variety of prior methods. Videos of policies learned by our algorithm can be viewed at https://sites.google.com/view/dnc-rl/.
+
+# 1 INTRODUCTION
+
+Deep reinforcement learning (RL) algorithms have demonstrated an impressive potential for tackling a wide range of complex tasks, from game playing (Mnih et al., 2015) to robotic manipulation (Levine et al., 2016; Kumar et al., 2016; Popov et al., 2017; Andrychowicz et al., 2017). However, many of the standard benchmark tasks in reinforcement learning, including the Atari benchmark suite (Mnih et al., 2013) and all of the OpenAI gym continuous control benchmarks (Brockman et al., 2016) lack the kind of diversity that is present in realistic environments.
+
+One of the most compelling use cases for RL algorithms is to create autonomous agents that can interact intelligently with diverse stochastic environments. However, such environments present a major challenge for current RL algorithms. Environments which require a lot of diversity can be expressed in the framework of RL as having a “wide” stochastic initial state distribution for the underlying Markov decision process. Highly stochastic initial state distributions lead to highvariance policy gradient estimates, which in turn hamper effective learning. Similarly, diversity and variability can also be incorporated by picking a wide distribution over goals.
+
+In this paper, we explore RL algorithms that are especially well-suited for tasks with a high degree of variability in both initial and goal states. We argue that a large class of practically interesting real-world problems fall into this category, but current RL algorithms are poorly equipped to handle them, as illustrated in our experimental evaluation. Our main observation is that, for tasks with a high degree of initial state variability, it is often much easier to obtain effective solutions to individual parts of the initial state space and then merge these solutions into a single policy, than to solve the entire task as a monolithic stochastic MDP. To that end, we can autonomously partition the state distribution into a set of distinct “slices,” and train a separate policy for each slice. For example, if we imagine the task of picking up a block with a robotic arm, different slices might correspond to different initial positions of the block. Similarly, for placing the block, different slices will correspond to the different goal positions. For each slice, the algorithm might train a different policy with a distinct strategy. As the training proceeds, we can gradually merge the distinct policies into a single global policy that succeeds in the entire space, by employing a combination of mutual KL-divergence constraints and supervised distillation.
+
+It may at first seem surprising that this procedure provides benefit. After all, if the final global policy can solve the entire task, then surely each local policy also has the representational capacity to capture a strategy that is effective on the entire initial state space. However, it is worth considering that a policy in a reinforcement learning algorithm must be able to represent not only the final optimal policy, but also all of the intermediate policies during learning. By decomposing these intermediate policies over the different slices of the initial state space, our method enables effective learning even on tasks with very diverse initial state and goal distributions. Since variation in the initial state distribution leads to high variance gradient estimates, this strategy also benefits from the fact that gradients can be better estimated in the local slices leading to accelerated learning. Intermediate supervised distillation steps help share information between the local policies which accelerates learning for slow learning policies, and helps policies avoid local optima.
+
+The main contribution of this paper is a reinforcement learning algorithm specifically designed for tasks with a high degree of diversity and variability. We term this approach as divide-and-conquer (DnC) reinforcement learning. Detailed empirical evaluation on a variety of difficult robotic manipulation and locomotion scenarios reveals that the proposed DnC algorithm substantially improves the performance over prior techniques.
+
+# 2 RELATED WORK
+
+Prior work has addressed reinforcement learning tasks requiring diverse behaviors, both in locomotion (Heess et al., 2017) and manipulation (Osa et al., 2016; Kober et al., 2012; Andrychowicz et al., 2017; Nair et al., 2017; Rajeswaran et al., 2017a). However, these methods typically make a number of simplifications, such as the use of demonstrations to help guide reinforcement learning (Osa et al., 2016; Kober et al., 2012; Nair et al., 2017; Rajeswaran et al., 2017a), or the use of a higher-level action representation, such as Cartesian end-effector control for a robotic arm (Osa et al., 2016; Kober et al., 2012; Andrychowicz et al., 2017; Nair et al., 2017). We show that the proposed DnC approach can solve manipulation tasks such as grasping and catching directly in the low-level torque action space without the need for demonstrations or a high-level action representation.
+
+The selection of benchmark tasks in this work is significantly more complex than those leveraging Cartesian position action spaces in prior work (Andrychowicz et al., 2017; Nair et al., 2017) or the relatively simple picking setup proposed by Popov et al. (2017), which consists of minimal task variation and a variety of additional shaping rewards. In the domain of locomotion, we show that our approach substantially outperforms the direct policy search method proposed by Heess et al. (2017). Curriculum learning has been applied to similar problems in reinforcement learning, with approaches that require the practitioner to design a sequence of progressively harder subsets of the initial state distribution, culminating in the original task (Asada et al., 1996; Karpathy & van de Panne, 2012). Our method allows for arbitrary decompositions, and we further show that DnC can work with automatically generated decompositions without human intervention.
+
+Our method is related to guided policy search (GPS) algorithms (Levine & Koltun, 2013; Mordatch & Todorov, 2014; Levine et al., 2016). These algorithms train several “local” policies by using a trajectory-centric reinforcement learning method, and a single “global” policy, typically represented by a deep neural network, which attempts to mimic the local policies. The local policies are constrained to the global policy, typically via a KL-divergence constraint. Our method also trains local policies, though the local policies are themselves represented by more flexible, nonlinear neural network policies. Use of neural network policies significantly improves the representational power of the individual controllers and facilitates the use of various off-the-shelf reinforcement learning algorithms. Furthermore, we constrain the various policies to one another, rather than to a single central policy, which we find substantially improves performance, as discussed in Section 5.
+
+Along similar lines, Teh et al. (2017) propose an approach similar to GPS for the purpose of transfer learning, where a single policy is trained to mimic the behavior of policies trained in specific domains. Our approach resembles GPS, in that we decompose a single complex task into local pieces, but also resembles Teh et al. (2017), in that we use nonlinear neural network local policies. Although Teh et al. (2017) propose a method intended for transfer learning, it can be adapted to the setting of stochastic initial states for comparison. We present results demonstrating that our approach substantially outperforms the method of Teh et al. (2017) in this setting.
+
+# 3 PRELIMINARIES
+
+An episodic Markov decision process (MDP) is defined as $\mathbf { M } = ( S , \mathcal { A } , P , r , \rho )$ where $s , A$ are continuous sets of states and actions respectively. $P ( s ^ { \prime } , s , a )$ is the transition probability distribution, $r : { \mathcal { S } } \mathbb { R }$ is the reward function, and $\rho : S \to \mathbb { R } _ { + }$ is the initial state distribution. We consider a modified MDP formulation, where the initial state distribution is conditioned on some variable $\omega$ , which we refer to as a “context.” Formally, $\Omega = ( \omega _ { i } ) _ { i = 1 } ^ { n }$ is a finite set of contexts, and $\rho : \Omega \times S $ $\mathbb { R } _ { + }$ is a joint distribution over contexts $\omega$ and initial states $s _ { 0 }$ . One can imagine that sampling initial states is a two stage process: first, contexts are sampled as $\rho ( \omega )$ , and then initial states are drawn given the sampled context as $\rho ( s | \omega )$ . Note that this formulation of a MDP with context is unrelated to the similarly named “contextual MDPs” (Hallak et al., 2015).
+
+For an arbitrary MDP, one can embed the context into the state as an additional state variable with independent distribution structure that factorizes as $P ( s _ { 0 } , \omega ) = P ( s _ { 0 } | \omega ) P ( \omega )$ . The context will provide us with a convenient mechanism to solve complex tasks, but we will describe how we can still train policies that, at convergence, no longer require any knowledge of the context, and operate only on the raw state of the MDP. We aim to find a stochastic policy $\pi : { \mathcal { S } } , { \mathcal { A } } \to \mathbb { R } _ { + }$ under which the expected reward of the policy $\eta ( \pi ) = \operatorname { \mathbb { E } } _ { \tau \sim \pi } [ r ( \tau ) ]$ is maximized.
+
+# 4 DIVIDE-AND-CONQUER REINFORCEMENT LEARNING
+
+In this section, we derive our divide-and-conquer reinforcement learning algorithm. We first motivate the approach by describing a policy learning framework for the MDP with context described above, and then introduce a practical algorithm that can implement this framework for complex reinforcement learning problems.
+
+# 4.1 LEARNING POLICIES FOR MDPS WITH CONTEXT
+
+We consider two extensions of the MDP M that exploit this contextual structure. First, we define an augmented MDP $\mathbf { M } ^ { \prime }$ that augments each state with information about the context $( \boldsymbol { S } \times \Omega , \boldsymbol { A } , P , \boldsymbol { r } , \rho )$ ; a trajectory in this MDP is $\bar { \tau = } ( ( \omega , s _ { 0 } ) , a _ { 0 } , ( \omega , s _ { 1 } ) , a _ { 1 } , \ldots )$ . We also consider the class of contextrestricted MDPs: for a context $\omega$ , we have $\mathbf { M } _ { \omega } = ( S , A , P , r , \rho _ { \omega } )$ , where $\rho _ { \omega } ( s ) = \mathbb { P } ( s | \Omega = \omega )$ ; i.e. the context is always fixed to $\omega$ .
+
+A stochastic policy $\pi$ in the augmented MDP $\mathbf { M } ^ { \prime }$ decouples into a family of simpler stochastic policies $\pi = ( \pi _ { i } ) _ { i = 1 } ^ { n }$ , where $\pi _ { i } : { \mathcal { S } } , { \mathcal { A } } \to [ 0 , 1 ]$ , and $\pi _ { i } ( s , a ) = \pi ( ( \omega _ { i } , s ) , a )$ . We can consider $\pi _ { i }$ to be a policy for the context-restricted MDP $\mathbf { M } _ { \omega _ { i } }$ , resulting in an equivalence between optimal policies in augmented MDPs and context-restricted MDPs. A family of optimal policies in the class of context-restricted MDPs is an optimal policy $\pi$ in $\mathbf { M } ^ { \prime }$ .This implies that policy search in the augmented MDP reduces to policy search in the context-restricted MDPs.
+
+Given a stochastic policy in the augmented MDP $( \pi _ { i } ) _ { i = 1 } ^ { n }$ , we can induce a stochastic policy $\pi _ { c }$ in the original MDP, by defining $\begin{array} { r } { \pi _ { c } ( s , a ) = \sum _ { \omega \in \Omega } p ( \omega | s ) \overline { { \pi _ { \omega } } } ( s , a ) } \end{array}$ , where $p ( \cdot | s )$ is a belief distribution of what context the trajectory is in. From here on, we refer to $\pi _ { c }$ as the central or global policy, and $\pi _ { i }$ as the context-specific or local policies.
+
+Our insight is that it is important for each local policy to not only be good for its designated context, but also be capable of working in other contexts. Requiring that local policies be capable of working broadly allows for sharing of information so that local policies designated for difficult contexts can bootstrap their solutions off easier contexts. As discussed in the previous section, we seek a policy in the original MDP, and local policies that generalize well to many other contexts induce global policies that are capable of operating in the original MDP, where no context is provided.
+
+In order to find the optimal policy for the original MDP, we search for a policy $\pi = ( \pi _ { i } ) _ { i = 1 } ^ { n }$ in the augmented MDP that maximizes $\eta ( \pi ) - \alpha \mathbb { E } _ { \pi } [ D _ { K L } ( \pi \| \pi _ { c } ) ] :$ maximizing expected reward for each instance while remaining close to a central policy, where $\alpha$ is a penalty hyperparameter. This encourages the central policy $\pi _ { c }$ to work for all the contexts, thereby transferring to the original
+
+MDP. Using Jensen’s inequality to bound the KL divergence between $\pi$ and $\pi _ { c }$ , we minimize the right hand side of Equation 1 as a bound for minimizing the intractable KL divergence optimization problem.
+
+$$
+\mathbb { E } _ { \pi } [ D _ { K L } ( \pi \| \pi _ { c } ) ] \leq \sum _ { i , j } \rho ( \omega _ { i } ) \rho ( \omega _ { j } ) \mathbb { E } _ { \pi _ { i } } [ D _ { K L } ( \pi _ { i } \| \pi _ { j } ) ]
+$$
+
+Equation 1 shows that finding a set of local policies that translates well into a global policy reduces into minimizing a weighted sum of pairwise KL divergence terms between local policies.
+
+# 4.2 THE DIVIDE-AND-CONQUER REINFORCEMENT LEARNING ALGORITHM
+
+We now present a policy optimization algorithm that takes advantage of this contextual starting state, following the framework discussed in the previous section. Given an MDP with structured initial state variation, we add contextual information by inducing contexts from a partition of the initial state distribution. More precisely, for a partition of $\textstyle S = \bigcup _ { i = 1 } ^ { n } S _ { i }$ , we associate a context $\omega _ { i }$ to each set $S _ { i }$ , so that $\omega = \omega _ { i }$ when $s _ { 0 } \in S _ { i }$ . This partition of the initial state space is generated by sampling initial states from the MDP, and running an automated clustering procedure. We use $\mathbf { k }$ -means clustering in our evaluation, since we focus on tasks with highly stochastic and structured initial states, and recommend alternative procedures for tasks with more intricate stochasticity.
+
+Having retrieved contexts $( \omega _ { i } ) _ { i = 1 } ^ { n }$ , we search for a global policy $\pi _ { c }$ by learning local policies $( \pi _ { i } ) _ { i = 1 } ^ { n }$ that maximize expected reward in the individual contexts, while constrained to not diverge from one another. We modify policy gradient algorithms, which directly optimize the parameters of a stochastic policy through local gradient-based methods, to optimize the local policies with our constraints.
+
+In particular, we base our algorithm on trust region policy optimization, TRPO, (Schulman et al., 2015), a policy gradient method which takes gradient steps according to the surrogate loss ${ \mathcal { L } } ( \pi )$ in Equation 2 while constraining the mean divergence from the old policy by a fixed constant.
+
+$$
+{ \mathcal { L } } ( \pi ) = - \mathbb { E } _ { \pi _ { o l d } } \left[ A ( s , a ) { \frac { \pi ( a | s ) } { \pi _ { o l d } ( a | s ) } } \right]
+$$
+
+We choose TRPO for its practical performance on high-dimensional continuous control problems, but our procedure extends easily to other policy gradient methods (Kakade, 2002; Williams, 1992) as well.
+
+In our framework, we optimize $\eta ( \pi ) - \alpha \mathbb { E } _ { \pi } [ D _ { K L } ( \pi \| \pi _ { c } ) ]$ , where $\alpha$ determines the relative balancing effect of expected reward and divergence. We adapt the TRPO surrogate loss to this objective, and with the bound in Equation 1, the surrogate objective simplifies to
+
+$$
+\mathcal { L } ( \pi _ { 1 } \dots \pi _ { n } ) = - \sum _ { i = 1 } ^ { n } \mathbb { E } _ { \pi _ { i , o l d } } \left[ A ( s , a ) \frac { \pi _ { i } ( a | s ) } { \pi _ { i , o l d } ( a | s ) } \right] + \alpha \left( \sum _ { i , j } \rho ( \omega _ { i } ) \rho ( \omega _ { j } ) \mathbb { E } _ { \pi _ { i } } \left[ D _ { K L } ( \pi _ { i } | | \pi _ { j } ) \right] \right)
+$$
+
+The KL divergence penalties encourage each local policy $\pi _ { i }$ to be close to other local policies on its own context $\omega _ { i }$ , and to mimic the other local policies on other contexts. As with standard policy gradients, the objective for $\pi _ { i }$ uses trajectories from context $\omega _ { i }$ , but the constraint on other contexts adds additional dependencies on trajectories from all the other contexts $( \omega _ { j } ) _ { j \neq i }$ . Despite only taking actions in a restricted context, each local policy is trained with data from the full context distribution. In Equation 4, we consider the loss as a function of a single local policy $\pi _ { i }$ , which reveals the dependence on data from the full context distribution, and explicitly lists the pairwise KL divergence penalties.
+
+$$
+\Sigma ( \pi ) \propto _ { \pi _ { i } } \underbrace { - \mathbb { E } _ { \pi _ { i , o l d } } \left[ A ( s , a ) \frac { \pi _ { i } ( a | s ) } { \pi _ { i , o l d } ( a | s ) } \right] } _ { \mathrm { M a x i m i z e s ~ } \eta ( \pi _ { i } ) } + \alpha \rho ( \omega _ { i } ) \sum _ { j } \rho ( \omega _ { j } ) \left( \underbrace { \mathbb { E } _ { \pi _ { i } } [ D _ { K L } ( \pi _ { i } | | \pi _ { j } ) ] } _ { \mathrm { C o n s t r a i n t o n ~ o n e n t e x t } } + \underbrace { \mathbb { E } _ { \pi _ { j } } [ D _ { K L } ( \pi _ { j } | | \pi _ { i } ) ] } _ { \mathrm { C o n s t r a i n t o n ~ o n e r ~ c o n e x t } } \right)
+$$
+
+On each iteration, trajectories from each context-restricted MDP $\mathbf { M } _ { \omega _ { i } }$ are collected using $\pi _ { i }$ , and each local policy $\pi _ { i }$ takes a gradient step with the surrogate loss in succession. It should be noted that the cost of evaluating and optimizing the KL divergence penalties grows quadratically with the number of contexts, as the number of penalty terms is quadratic. For practical tasks however, the number of contexts will be on the order of 5-10, and the quadratic cost imposes minimal overhead for the TRPO conjugate gradient evaluation.
+
+After repeating the trajectory collection and policy optimization procedure for a fixed number of iterations, we seek to retrieve $\pi _ { c }$ , a central policy in the original task from the local policies trained via TRPO. As discussed in Section 4.1, this corresponds to minimizing the KL divergence between $\pi$ and $\pi _ { c }$ , which neatly simplifies into a maximum likelihood problem with samples from all the various policies.
+
+$$
+\mathcal { L } _ { c e n t e r } ( \pi _ { c } ) = \mathbb { E } _ { \pi } [ D _ { K L } ( \pi ( \cdot | s ) \| \pi _ { c } ( \cdot | s ) ) ] \propto \sum _ { i } \rho ( \omega _ { i } ) \mathbb { E } _ { \pi _ { i } } \left[ - \log \pi _ { c } ( s , a ) \right]
+$$
+
+If $\pi _ { c }$ performs inadequately on the full task, the local policy training procedure is repeated, initializing the local policies to start at $\pi _ { c }$ . We alternately optimize the local policies and the global policy in this manner, until convergence. The algorithm is laid out fully in pseudocode below.
+
+| R ←Distillation Period function DNC() |
| Sample initial states so from the task |
| Produce contexts W1, W2,...Wn by clustering initial states so |
| Randomly initialize central policy Tc |
| for t = 1,2... until convergence do |
| Setπi=πc foralli=1...n |
| for R iterations do |
| Collect trajectories Ti in context wi using policy πi for all i = 1...n for all local policies Ti do |
| Take gradient step in surrogate loss L wrt Ti |
| Minimize Lcenter W.r.t. πc using previously sampled states (Ti)=1 |
| return Tc |
+
+# 5 EXPERIMENTAL EVALUATION
+
+We focus our analysis on tasks spanning two different domains: manipulation and locomotion. Manipulation tasks involve handling an un-actuated object with a robotic arm, and locomotion tasks involve tackling challenging terrains. We illustrate a variety of behaviors in both settings. Standard continuous control benchmarks are known to represent relatively mild representational challenges (Rajeswaran et al., 2017b), and thus it was important to design new tasks that are more challenging in order to illustrate the potential of proposed approach. Tasks were designed to bring out complex contact rich behaviors in settings with considerable variation and diversity. All of our environments are designed and simulated in MuJoCo (Todorov et al., 2012).
+
+Our experiments and analysis aim to address the following questions:
+
+1. Can DnC solve highly complex tasks in a variety of domains, especially tasks that cannot be solved with current conventional policy gradient methods? 2. How does the form of the constraint on the ensemble policies in DnC affect the performance of the final policy, as compared to previously proposed constraints?
+
+We compare DnC to the following prior methods and ablated variants:
+
+• TRPO. TRPO (Schulman et al., 2015) represents a state-of-the-art policy gradient method, which we use for the standard RL comparison without decomposition into contexts. TRPO is provided with the same batch size as the sum of the batches over all of the policies in our algorithm, to ensure a fair comparison. Distral. Originally formulated as transfer learning in a discrete action space (Teh et al., 2017), we extend Distral to our stochastic initial state continuous control setting, where each context $\omega$ is a different task. For proper comparison between the algorithms, we port
+
+Distral to the TRPO objective, since empirically TRPO outperforms other policy gradient methods in this domain. This algorithm, which resembles the structure of guided policy search, also trains an ensemble of policies, but constrains them at each gradient step against a single global policy trained with supervised learning, and omits the distillation step that our method performs every $R$ iterations.
+
+• Unconstrained DnC. We run the DnC algorithm without any KL constraints. This reduces to running TRPO to train policies $( \pi _ { i } ) _ { i = 1 } ^ { n }$ on each context, and distilling the resulting local policies every $R$ iterations.
+
+• Centralized DnC. Whereas DnC doesn’t perform inference on context, centralized DnC uses an oracle to perfectly identify the context $\omega$ from the state $s$ . The resulting algorithm is equivalent to the Distral objective, but distills every $R$ steps.
+
+Performance of these methods is highly dependent on the choice of $\alpha$ , the penalty hyperparameter that controls how tightly to couple the local policies. For each task, we run a hyperparameter sweep for each method, showing results for the best penalty weight. Furthermore, performance of policy gradient methods like TRPO varies significantly from run to run, so we run each experiment with 5 random seeds, reporting mean statistics and standard deviations. The experimental procedure is detailed more extensively in Appendix A.
+
+For each evaluation task, we use a k-means clustering procedure to partition the initial state space into four contexts, which we found empirically to create stable partitions, and yield high performance across algorithms. We further detail the clustering procedure and examine the effect of partition size on DnC in Appendix C. The focus of our work is finding a single global policy that performs well on the full state space, but we further compare to oracle-based ensemble policies in Appendix D.
+
+# 6 ROBOTIC MANIPULATION
+
+For robotic manipulation, we simulate the Kinova Jaco, a 7 DoF robotic arm with 3 fingers. The agent receives full state information, which includes the current absolute location of external objects such as boxes. The agent uses low-level joint torque control to perform the required actions. Note that use of low-level torque control significantly increases complexity, as raw torque control on a 7 DoF arm requires delicate movements of the joints to perform each task. We describe the tasks below, and present full specifics in Appendix B.
+
+Picking. The Picking task requires the Jaco to pick up a small block and raise it as high as possible. The agent receives reward only when the block is in the agent’s hand. The starting position of the block is randomized within a fixed $3 0 \mathrm { c m }$ by $3 0 \mathrm { c m }$ square surface on the table. Picking up the block from different locations within the workspace require diverse poses, making this task challenging in the torque control framework. TRPO can only solve the picking task from a 4cm by 4cm workspace, and from wider configurations, the Jaco fails to grasp with a high success rate with policies learnt via TRPO.
+
+Lobbing. The Lobbing task requires the Jaco to flick a block into a target box, which is placed in a randomized location within a 1m by 1m square, far enough that the arm cannot reach it directly. This problem inherits many challenges from the picking task. Furthermore, the sequential nature of grasping and flicking necessitates that information pass temporally and requires synthesis of multiple skills.
+
+Catching. In the Catching task, a ball is thrown at the robot with randomized initial position and velocity, and the arm must catch it in the air. Fixed reward is awarded every step that the ball is in or next to the hand. This task is particularly challenging due the temporal sensitivity of the problem, since the end-effector needs to be in perfect sync with the flying object successfully finish the grasp. This extreme temporal dependency renders stochastic estimates of the gradients ineffective in guiding the learning.
+
+DnC exceeds the performance of the alternative methods on all of the manipulation tasks, as shown in Figure 1. For each task, we include the average reward, as well as a success rate measure, which provides a more interpretable impression of the performance of the final policy. TRPO by itself is unable to solve any of the tasks, with success rates below $10 \%$ in each case. The policies learned by
+
+
+Figure 1: Average return and success rate learning curves of the global policy on Picking, Lobbing, Catching, Ant, and Stairs when partitioned into 4 contexts. Metrics are evaluated each iteration on the global policy distilled from the current local policies at that iteration. On all of the tasks, DnC RL achieves the best results. On the Catching and Ant tasks, DnC performs comparably to the centralized variant, while on the Picking, Lobbing, and Stairs tasks, the full algorithm outperforms all others by a wide margin. All of the experiments are shown with 5 random seeds.
+
+| R ←Distillation Period function DNC() |
| Sample initial states so from the task |
| Produce contexts W1, W2,...Wn by clustering initial states so |
| Randomly initialize central policy Tc |
| for t = 1,2... until convergence do |
| Setπi=πc foralli=1...n |
| for R iterations do |
| Collect trajectories Ti in context wi using policy πi for all i = 1...n for all local policies Ti do |
| Take gradient step in surrogate loss L wrt Ti |
| Minimize Lcenter W.r.t. πc using previously sampled states (Ti)=1 |
| return Tc |
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+ "text": "• TRPO. TRPO (Schulman et al., 2015) represents a state-of-the-art policy gradient method, which we use for the standard RL comparison without decomposition into contexts. TRPO is provided with the same batch size as the sum of the batches over all of the policies in our algorithm, to ensure a fair comparison. Distral. Originally formulated as transfer learning in a discrete action space (Teh et al., 2017), we extend Distral to our stochastic initial state continuous control setting, where each context $\\omega$ is a different task. For proper comparison between the algorithms, we port ",
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+ "text": "Distral to the TRPO objective, since empirically TRPO outperforms other policy gradient methods in this domain. This algorithm, which resembles the structure of guided policy search, also trains an ensemble of policies, but constrains them at each gradient step against a single global policy trained with supervised learning, and omits the distillation step that our method performs every $R$ iterations. ",
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+ "text": "6 ROBOTIC MANIPULATION ",
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+ "text": "For robotic manipulation, we simulate the Kinova Jaco, a 7 DoF robotic arm with 3 fingers. The agent receives full state information, which includes the current absolute location of external objects such as boxes. The agent uses low-level joint torque control to perform the required actions. Note that use of low-level torque control significantly increases complexity, as raw torque control on a 7 DoF arm requires delicate movements of the joints to perform each task. We describe the tasks below, and present full specifics in Appendix B. ",
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+ "text": "Picking. The Picking task requires the Jaco to pick up a small block and raise it as high as possible. The agent receives reward only when the block is in the agent’s hand. The starting position of the block is randomized within a fixed $3 0 \\mathrm { c m }$ by $3 0 \\mathrm { c m }$ square surface on the table. Picking up the block from different locations within the workspace require diverse poses, making this task challenging in the torque control framework. TRPO can only solve the picking task from a 4cm by 4cm workspace, and from wider configurations, the Jaco fails to grasp with a high success rate with policies learnt via TRPO. ",
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+ "text": "DnC exceeds the performance of the alternative methods on all of the manipulation tasks, as shown in Figure 1. For each task, we include the average reward, as well as a success rate measure, which provides a more interpretable impression of the performance of the final policy. TRPO by itself is unable to solve any of the tasks, with success rates below $10 \\%$ in each case. The policies learned by ",
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+ "Figure 1: Average return and success rate learning curves of the global policy on Picking, Lobbing, Catching, Ant, and Stairs when partitioned into 4 contexts. Metrics are evaluated each iteration on the global policy distilled from the current local policies at that iteration. On all of the tasks, DnC RL achieves the best results. On the Catching and Ant tasks, DnC performs comparably to the centralized variant, while on the Picking, Lobbing, and Stairs tasks, the full algorithm outperforms all others by a wide margin. All of the experiments are shown with 5 random seeds. "
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+ "table_body": "| R ←Distillation Period function DNC() |
| Sample initial states so from the task |
| Produce contexts W1, W2,...Wn by clustering initial states so |
| Randomly initialize central policy Tc |
| for t = 1,2... until convergence do |
| Setπi=πc foralli=1...n |
| for R iterations do |
| Collect trajectories Ti in context wi using policy πi for all i = 1...n for all local policies Ti do |
| Take gradient step in surrogate loss L wrt Ti |
| Minimize Lcenter W.r.t. πc using previously sampled states (Ti)=1 |
| return Tc |
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| Sample initial states so from the task |
| Produce contexts W1, W2,...Wn by clustering initial states so |
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| for t = 1,2... until convergence do |
| Setπi=πc foralli=1...n |
| for R iterations do |
| Collect trajectories Ti in context wi using policy πi for all i = 1...n for all local policies Ti do |
| Take gradient step in surrogate loss L wrt Ti |
| Minimize Lcenter W.r.t. πc using previously sampled states (Ti)=1 |
| return Tc |
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+ "html": "| Name | Num Agents | Policy Representation | Part of State = (v,t,d) Used |
| routing-all-fc | Num Tasks | Tabular (numlayers X num function blocks) | t,d |
| routing-all-fc non-layered | Num Tasks | tabular (num layers X num function blocks) | t,d |
| soft-routing-all-fc | Num Tasks | Vector(num layers) of appox functions | v,t,d |
| dispatched-routing-all-fc | Num Tasks +1 | Vector (num layers) of appox functions+ dispatcher | v,td |
| single-agent-routing-all-fc | 1 | Vector (num layers)of approx functions) | v,t,d |
+
+Policy Representation There are two dominant representation variations, as described in 3.1. In the first, the policy is stored as a table. Since the table needs to store values for each of the different layers of the routing network, it is of size num layers $\times$ num actions. In the second, it is represented either as vector of MLP’s with a hidden layer of dimension 64, one for each layer of the routing network. In this case the input to the MLP is the representation vector $v$ concatenated with a one-hot representation of the task identifier.
+
+Policy Input describes which parts of the state are used in the decision of the routing action. For tabular policies, the task is used to index the agent responsible for handling that task. Each agent then uses the depth as a row index into into the table. For approximation-based policies, there are two variations. For the single agent case the depth is used to index an approximation function which takes as input concat $_ v$ , one-hot $\mathbf { \eta } ^ { ( t ) }$ ). For the multi-agent (non-dispatched) case the task label is used to index the agent and then the depth is used to index the corresponding approximation function for that depth, which is given concat(v, one-hot $\mathbf { \rho } ( t )$ ) as input. In the dispatched case, the dispatcher is given concat $\dot { \boldsymbol { v } }$ , one-hot(t)) and predicts an agent index. That agent uses the depth to find the approximation function for that depth which is then given concat( $\boldsymbol { v }$ , one-hot $\mathbf { \Psi } ( t ) .$ ) as input.
+
+# 7.4 EXPLANATION OF THE WEIGHTED POLICY LEARNER (WPL) ALGORITHM
+
+The WPL algorithm is a multi-agent policy gradient algorithm designed to help dampen policy oscillation and encourage convergence. It does this by slowly scaling down the learning rate for an agent after a gradient change in that agents policy. It determines when there has been a gradient change by using the difference between the immediate reward and historical average reward for the action taken. Depending on the sign of the gradient the algorithm is in one of two scenarios. If the gradient is positive then it is scaled by $1 - \pi ( a _ { i } )$ . Over time if the gradient remains positive it will cause $\pi ( \boldsymbol { a } _ { i } )$ to increase and so $1 - \pi ( a _ { i } )$ will go to 0, slowing the learning. If the gradient is negative then it is scaled by $\pi ( \boldsymbol { a } _ { i } )$ . Here again if the gradient remains negative over time it will cause $\pi ( \boldsymbol { a } _ { i } )$ to decrease eventually to 0, slowing the learning again. Slowing the learning after gradient changes dampens the policy oscillation and helps drive the policies towards convergence.
\ No newline at end of file
diff --git a/parse/train/ry8dvM-R-/ry8dvM-R-_content_list.json b/parse/train/ry8dvM-R-/ry8dvM-R-_content_list.json
new file mode 100644
index 0000000000000000000000000000000000000000..4f24ebf1d27e5e70ba5badaaf87cc92cc650e64b
--- /dev/null
+++ b/parse/train/ry8dvM-R-/ry8dvM-R-_content_list.json
@@ -0,0 +1,1733 @@
+[
+ {
+ "type": "text",
+ "text": "ROUTING NETWORKS: ADAPTIVE SELECTION OF NON-LINEAR FUNCTIONS FOR MULTI-TASK LEARNING ",
+ "text_level": 1,
+ "bbox": [
+ 174,
+ 98,
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+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Clemens Rosenbaum \nCollege of Information and Computer Sciences \nUniversity of Massachusetts Amherst \n140 Governors Dr., Amherst, MA 01003 \ncgbr@cs.umass.edu ",
+ "bbox": [
+ 184,
+ 195,
+ 493,
+ 265
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Tim Klinger & Matthew Riemer IBM Research AI 1101 Kitchawan Rd, Yorktown Heights, NY 10598 {tklinger,mdriemer}@us.ibm.com ",
+ "bbox": [
+ 509,
+ 195,
+ 846,
+ 252
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "ABSTRACT ",
+ "text_level": 1,
+ "bbox": [
+ 454,
+ 301,
+ 544,
+ 316
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Multi-task learning (MTL) with neural networks leverages commonalities in tasks to improve performance, but often suffers from task interference which reduces the benefits of transfer. To address this issue we introduce the routing network paradigm, a novel neural network and training algorithm. A routing network is a kind of self-organizing neural network consisting of two components: a router and a set of one or more function blocks. A function block may be any neural network – for example a fully-connected or a convolutional layer. Given an input the router makes a routing decision, choosing a function block to apply and passing the output back to the router recursively, terminating when a fixed recursion depth is reached. In this way the routing network dynamically composes different function blocks for each input. We employ a collaborative multi-agent reinforcement learning (MARL) approach to jointly train the router and function blocks. We evaluate our model against cross-stitch networks and shared-layer baselines on multi-task settings of the MNIST, mini-imagenet, and CIFAR-100 datasets. Our experiments demonstrate a significant improvement in accuracy, with sharper convergence. In addition, routing networks have nearly constant per-task training cost while cross-stitch networks scale linearly with the number of tasks. On CIFAR100 (20 tasks) we obtain cross-stitch performance levels with an $8 5 \\%$ reduction in training time. ",
+ "bbox": [
+ 233,
+ 332,
+ 764,
+ 595
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "1 INTRODUCTION ",
+ "text_level": 1,
+ "bbox": [
+ 176,
+ 621,
+ 336,
+ 637
+ ],
+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Multi-task learning (MTL) is a paradigm in which multiple tasks must be learned simultaneously. Tasks are typically separate prediction problems, each with their own data distribution. In an early formulation of the problem, (Caruana, 1997) describes the goal of MTL as improving generalization performance by “leveraging the domain-specific information contained in the training signals of related tasks.” This means a model must leverage commonalities in the tasks (positive transfer) while minimizing interference (negative transfer). In this paper we propose a new architecture for MTL problems called a routing network, which consists of two trainable components: a router and a set of function blocks. Given an input, the router selects a function block from the set, applies it to the input, and passes the result back to the router, recursively up to a fixed recursion depth. If the router needs fewer iterations then it can decide to take a PASS action which leaves the current state unchanged. Intuitively, the architecture allows the network to dynamically self-organize in response to the input, sharing function blocks for different tasks when positive transfer is possible, and using separate blocks to prevent negative transfer. ",
+ "bbox": [
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+ "text": "The architecture is very general allowing many possible router implementations. For example, the router can condition its decision on both the current activation and a task label or just one or the other. It can also condition on the depth (number of router invocations), filtering the function module choices to allow layering. In addition, it can condition its decision for one instance on what was historically decided for other instances, to encourage re-use of existing functions for improved compression. The function blocks may be simple fully-connected neural network layers or whole networks as long as the dimensionality of each function block allows composition with the previous function block choice. They needn’t even be the same type of layer. Any neural network or part of a network can be “routed” by adding its layers to the set of function blocks, making the architecture applicable to a wide range of problems. Because the routers make a sequence of hard decisions, which are not differentiable, we use reinforcement learning (RL) to train them. We discuss the training algorithm in Section 3.1, but one way we have modeled this as an RL problem is to create a separate RL agent for each task (assuming task labels are available in the dataset). Each such task agent learns its own policy for routing instances of that task through the function blocks. ",
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+ "text": "To evaluate we have created a “routed” version of the convnet used in (Ravi & Larochelle, 2017) and use three image classification datasets adapted for MTL learning: a multi-task MNIST dataset that we created, a Mini-imagenet data split as introduced in (Vinyals et al., 2016), and CIFAR-100 (Krizhevsky, 2009), where each of the 20 label superclasses are treated as different tasks.1 We conduct extensive experiments comparing against cross-stitch networks (Misra et al., 2016) and the popular strategy of joint training with layer sharing as described in (Caruana, 1997). Our results indicate a significant improvement in accuracy over these strong baselines with a speedup in convergence and often orders of magnitude improvement in training time over cross-stitch networks. ",
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+ "text": "2 RELATED WORK ",
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+ "text": "Work on multi-task deep learning (Caruana, 1997) traditionally includes significant hand design of neural network architectures, attempting to find the right mix of task-specific and shared parameters. For example, many architectures share low-level features like those learned in shallow layers of deep convolutional networks or word embeddings across tasks and add task-specific architectures in later layers. By contrast, in routing networks, we learn a fully dynamic, compositional model which can adjust its structure differently for each task. ",
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+ "text": "Routing networks share a common goal with techniques for automated selective transfer learning using attention (Rajendran et al., 2017) and learning gating mechanisms between representations (Stollenga et al., 2014), (Misra et al., 2016), (Ruder et al., 2017). In the latter two papers, experiments are performed on just 2 tasks at a time. We consider up to 20 tasks in our experiments and compare directly to (Misra et al., 2016). ",
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+ "text": "Our work is also related to mixtures of experts architectures (Jacobs et al., 1991), (Jordan & Jacobs, 1994) as well as their modern attention based (Riemer et al., 2016) and sparse (Shazeer et al., 2017) variants. The gating network in a typical mixtures of experts model takes in the input and chooses an appropriate weighting for the output of each expert network. This is generally implemented as a soft mixture decision as opposed to a hard routing decision, allowing the choice to be differentiable. Although the sparse and layer-wise variant presented in (Shazeer et al., 2017) does save some computational burden, the proposed end-to-end differentiable model is only an approximation and doesn’t model important effects such as exploration vs. exploitation tradeoffs, despite their impact on the system. Mixtures of experts have recently been considered in the transfer learning setting (Aljundi et al., 2016), however, the decision process is modelled by an autoencoder-reconstructionerror-based heuristic and is not scaled to a large number of tasks. ",
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+ "text": "In the use of dynamic representations, our work is also related to single task and multi-task models that learn to generate weights for an optimal neural network (Ha et al., 2016), (Ravi & Larochelle, 2017), (Munkhdalai & Yu, 2017). While these models are very powerful, they have trouble scaling to deep models with a large number of parameters (Wichrowska et al., 2017) without tricks to simplify the formulation. In contrast, we demonstrate that routing networks can be applied to create dynamic network architectures for architectures like convnets by routing some of their layers. ",
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+ "text": "Our work extends an emerging line of recent research focused on automated architecture search. In this work, the goal is to reduce the burden on the practitioner by automatically learning black box algorithms that search for optimal architectures and hyperparameters. These include techniques based on reinforcement learning (Zoph & Le, 2017), (Baker et al., 2017), evolutionary algorithms (Miikkulainen et al., 2017), approximate random simulations (Brock et al., 2017), and adaptive growth (Cortes et al., 2016). To the best of our knowledge we are the first to apply this idea to multitask learning. Our technique can learn to construct a very general class of architectures without the need for human intervention to manually choose which parameters will be shared and which will be kept task-specific. ",
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+ "text": "Also related to our work is the literature on minimizing computation cost for single-task problems by conditional routing. These include decisions trained with REINFORCE (Denoyer & Gallinari, 2014), (Bengio et al., 2015), (Hamrick et al., 2017), Q Learning (Liu & Deng, 2017), and actor-critic methods (McGill & Perona, 2017). Our approach differs however in the introduction of several novel elements. Specifically, our work explores the multi-task learning setting, it uses a multi-agent reinforcement learning training algorithm, and it is structured as a recursive decision process. ",
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+ "text": "There is a large body of related work which focuses on continual learning, in which tasks are presented to the network one at a time, potentially over a long period of time. One interesting recent paper in this setting, which also uses the notion of routes (“paths”), but uses evolutionary algorithms instead of RL is Fernando et al. (2017). ",
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+ "text": "While a routing network is a novel artificial neural network formulation, the high-level idea of task specific “routing” as a cognitive function is well founded in biological studies and theories of the human brain (Gurney et al., 2001), (Buschman & Miller, 2010), (Stocco et al., 2010). ",
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+ "text": "3 ROUTING NETWORKS",
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+ "Figure 1: Routing (forward) Example "
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+ "text": "A routing network consists of two components: a router and a set of function blocks, each of which can be any neural network layer. The router is a function which selects from among the function blocks given some input. Routing is the process of iteratively applying the router to select a sequence of function blocks to be composed and applied to the input vector. This process is illustrated in Figure 1. The input to the routing network is an instance to be classified $( v , \\dot { t } ) , v \\in \\mathbb { R } ^ { d }$ is a representation vector of dimension $d$ and $t$ is an integer task identifier. The router is given $v , t$ and a depth $( = 1 )$ , the depth of the recursion, and selects from among a set of function block choices available at depth 1, $\\left\\{ f _ { 1 3 } , f _ { 1 2 } , f _ { 1 1 } \\right\\}$ , picking $f _ { 1 3 }$ which is indicated with a dashed line. $f _ { 1 3 }$ is applied to the input $( v , t )$ to produce an output activation. The router again chooses a function block from those available at depth 2 (if the function blocks are of different dimensions then the router is constrained to select dimensionally matched blocks to apply) and so on. Finally the router chooses a function block from the last (classification) layer function block set and produces the classification $\\hat { y }$ . ",
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+ "text": "Algorithm 1 gives the routing procedure in detail. The algorithm takes as input a vector $v$ , task label $t$ and maximum recursion depth $n$ . It iterates $n$ times choosing a function block on each iteration and applying it to produce an output representation vector. A special PASS action (see Appendix Section 7.2 for details) just skips to the next iteration. Some experiments don’t require a task label and in that case we just pass a dummy value. For simplicity we assume the algorithm has access to the router function and function blocks and don’t include them explicitly in the input. The router decision function router : $\\mathbb { R } ^ { d } \\times \\mathbb { Z } ^ { + } \\times \\mathbb { Z } ^ { + } $ $\\{ 1 , 2 , \\ldots , k , P A S S \\}$ (for $d$ the input representation dimension and $k$ the number of function blocks) maps the current representation $v$ , task label $t \\in \\mathbb { Z } ^ { + }$ , and current depth $\\bar { i } \\in \\mathbb { Z } ^ { + }$ to the index of the function block to route next in the ordered set function block. ",
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+ "text": "Algorithm 1: Routing Algorithm ",
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+ "text": "input : $x , t , n$ : $x \\in \\mathbb { R } ^ { d }$ , $d$ the representation dim; $t$ integer task id; $n$ max depth output: $v$ - the vector result of applying the composition of the selected functions to the input $x$ 1 $v x$ 2 for $i$ in $1 . . . n$ do 3 $\\boldsymbol { a } \\gets \\mathbf { r o u t e r } ( \\boldsymbol { x } , t , i )$ 4 if $a \\neq P A S S$ then 5 $x \\gets$ function blocka(x) 6 return $\\nu$ ",
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+ "text": "If the routing network is run for $d$ invocations then we say it has depth $d$ . For $N$ function blocks a routing network run to a depth $d$ can select from $N ^ { d }$ distinct trainable functions (the paths in the network). Any neural network can be represented as a routing network by adding copies of its layers as routing network function blocks. We can group the function blocks for each network layer and constrain the router to pick from layer 0 function blocks at depth 0, layer 1 blocks at depth 1, and so on. If the number of function blocks differs from layer to layer in the original network, then the router may accommodate this by, for example, maintaining a separate decision function for each depth. ",
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+ "text": "3.1 ROUTER TRAINING USING RL ",
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+ "text": "Algorithm 2: Router-Trainer: Training of a Routing Network. ",
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+ "text": "input: A dataset $D$ of samples $( v , t , y )$ , $v$ the input representation, $t$ an integer task label, $y$ a ground-truth target label 1 for each sample $s = ( v , t , y ) \\in D$ do 2 Do a forward pass through the network, applying Algorithm 1 to sample $s$ . Store a trace $T = ( S , A , R , r _ { f i n a l } )$ , where $S =$ sequence of visited states $\\left( { { s } _ { i } } \\right)$ ; $A =$ sequence of actions taken $\\left( { { a } _ { i } } \\right)$ ; $R =$ sequence of immediate action rewards $( r _ { i } )$ for action $a _ { i }$ ; and the final reward rf inal. The last output as the network’s prediction $\\hat { y }$ and the final reward $r _ { f i n a l }$ is $+ 1$ if the prediction $\\hat { y }$ is correct; $^ { - 1 }$ if not. 3 Compute the loss $\\mathcal { L } ( \\hat { y } , y )$ between prediction $\\hat { y }$ and ground truth $y$ and backpropagate along the function blocks on the selected route to train their parameters. 4 Use the trace $T$ to train the router using the desired RL training algorithm. ",
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+ "text": "We can view routing as an RL problem in the following way. The states of the MDP are the triples $( v ,$ $t , i )$ where $v \\in \\mathbb { R } ^ { d }$ is a representation vector (initially the input), $t$ is an integer task label for $v$ , and $i$ is the depth (initially 1). The actions are function block choices (and PASS) in $\\{ 1 , \\dots k , P A S S \\}$ for $k$ the number of function blocks. Given a state $s = ( \\boldsymbol { v } , t , i )$ , the router makes a decision about which action to take. For the non-PASS actions, the state is then updated $s ^ { \\prime } = ( v ^ { \\prime } , t , i + 1 )$ and the process continues. The PASS action produces the same representation vector again but increments the depth, so $s ^ { \\prime } = ( v , t , i + 1 )$ . We train the router policy using a variety of RL algorithms and settings which we will describe in detail in the next section. ",
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+ "text": "Regardless of the RL algorithm applied, the router and function blocks are trained jointly. For each instance we route the instance through the network to produce a prediction $\\hat { y }$ . Along the way we record a trace of the states $s _ { i }$ and the actions $a _ { i }$ taken as well as an immediate reward $r _ { i }$ for action $a _ { i }$ . When the last function block is chosen, we record a final reward which depends on the prediction $\\hat { y }$ and the true label $y$ . ",
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+ "text": "$$\n\\begin{array} { r l r } & { \\left[ \\frac { \\hat { \\rho } _ { 1 3 } ^ { \\varDelta } } { \\int \\ d _ { 1 3 } } \\right] \\times \\frac { \\hat { \\rho } _ { 2 3 } ^ { \\varDelta } } { - \\left( - \\frac { 1 } { 2 } - 1 \\right) ! } = \\frac { \\hat { \\rho } _ { 2 2 } ^ { \\varDelta } } { - \\left( - \\frac { 1 } { 2 } - 1 \\right) ! } - \\left[ \\frac { \\hat { \\rho } _ { 3 2 } ^ { \\varDelta } } { 2 - 1 } - \\mathcal { L } ( \\hat { y } , y ) \\right] \\stackrel { \\mathrm { R o u t i n g ~ E x a m p l e ~ ( s e e ~ F i g u r e ~ 1 ) } } { - \\left( \\frac { 1 } { \\sqrt { 3 } } \\right) ! } \\Longrightarrow } & \\\\ & a _ { 1 } \\ll - \\frac { + r _ { 1 } } { - \\left( - \\frac { 1 } { 2 } - 1 - a _ { 2 } \\right) + \\left( - \\frac { 1 } { 2 } - \\frac { 1 } { 2 } - 1 - a _ { 3 } \\right) + \\left( \\frac { + r _ { 3 } } { 2 } - \\frac { 1 } { \\sqrt { 3 } } - r _ { f i n a l } \\right. _ { + } . . . . . . . . . . . . . . . . } \\end{array}\n$$",
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+ "Figure 2: Training (backward) Example "
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+ "text": "We train the selected function blocks using SGD/backprop. In the example of Figure 1 this means computing gradients for $f _ { 3 2 }$ , $f _ { 2 1 }$ and $f _ { 1 3 }$ . We then use the computed trace to train the router using an RL algorithm. The high-level procedure is summarized in Algorithm 2 and illustrated in Figure 2. To keep the presentation uncluttered we assume the RL training algorithm has access to the router function, function blocks, loss function, and any specific hyper-parameters such as discount rate needed for the training and don’t include them explicitly in the input. ",
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+ "text": "A routing network uses two kinds of rewards: immediate action rewards $r _ { i }$ given in response to an action $a _ { i }$ and a final reward $r _ { f i n a l }$ , given at the end of the routing. The final reward is a function of the network’s performance. For the classification problems focused on in this paper, we set it to $+ 1$ if the prediction was correct $( \\hat { y } = y )$ ), and $- 1$ otherwise. For other domains, such as regression domains, the negative loss $( - \\hat { \\cal L } ( \\hat { y } , y ) )$ could be used. ",
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+ "text": "We experimented with an immediate reward that encourages the router to use fewer function blocks when possible. Since the number of function blocks per-layer needed to maximize performance is not known ahead of time (we just take it to be the same as the number of tasks), we wanted to see whether we could achieve comparable accuracy while reducing the number of function blocks ever chosen by the router, allowing us to reduce the size of the network after training. We experimented with two such rewards, multiplied by a hyper-parameter $\\rho \\in [ 0 , 1 ]$ : the average number of times that block was chosen by the router historically and the average historical probability of the router choosing that block. We found no significant difference between the two approaches and use the average probability in our experiments. We evaluated the effect of $\\rho$ on final performance and report the results in Figure 12 in the appendix. We see there that generally $\\rho = 0 . 0$ (no collaboration reward) or a small value works best and that there is relatively little sensitivity to the choice in this range. ",
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+ "Figure 3: Task-based routing. $\\langle v a l u e , t a s k \\rangle$ is the input consisting of value, the partial evaluation of the previous function block (or input $x$ ) and the task label task. $\\alpha _ { i }$ is a routing agent; $\\alpha _ { d }$ is a dispatching agent. "
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+ "text": "To train the router we evaluate both single-agent and multi-agent RL strategies. Figure 3 shows three variations which we consider. In Figure 3(a) there is just a single agent which makes the routing decision. This is be trained using either policy-gradient (PG) or Q-Learning experiments. Figure 3(b) shows a multi-agent approach. Here there are a fixed number of agents and a hard rule which assigns the input instance to a an agent responsible for routing it. In our experiments we create one agent per task and use the input task label as an index to the agent responsible for routing that instance. Figure 3(c) shows a multi-agent approach in which there is an additional agent, denoted $\\alpha _ { d }$ and called a dispatching agent which learns to assign the input to an agent, instead of using a fixed rule. For both of these multi-agent scenarios we additionally experiment with a MARL algorithm called Weighted Policy Learner (WPL). ",
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+ "text": "We experiment with storing the policy both as a table and in form of an approximator. The tabular representation has the invocation depth as its row dimension and the function block as its column dimension with the entries containing the probability of choosing a given function block at a given depth. The approximator representation can consist of either one MLP that is passed the depth (represented in 1-hot), or a vector of $d$ MLPs, one for each decision/depth. ",
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+ "text": "Both the Q-Learning and Policy Gradient algorithms are applicable with tabular and approximation function policy representations. We use REINFORCE (Williams, 1992) to train both the approximation function and tabular representations. For Q-Learning the table stores the Q-values in the entries. We use vanilla Q-Learning (Watkins, 1989) to train tabular representation and train the approximators to minimize the $\\ell _ { 2 }$ norm of the temporal difference error. ",
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+ "text": "Implementing the router decision policy using multiple agents turns the routing problem into a stochastic game, which is a multi-agent extension of an MDP. In stochastic games multiple agents interact in the environment and the expected return for any given policy may change without any action on that agent’s part. In this view incompatible agents need to compete for blocks to train, since negative transfer will make collaboration unattractive, while compatible agents can gain by sharing function blocks. The agent’s (locally) optimal policies will correspond to the game’s Nash equilibrium 2 ",
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+ "text": "For routing networks, the environment is non-stationary since the function blocks are being trained as well as the router policy. This makes the training considerably more difficult than in the singleagent (MDP) setting. We have experimented with single-agent policy gradient methods such as REINFORCE but find they are less well adapted to the changing environment and changes in other agent’s behavior, which may degrade their performance in this setting. ",
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+ "text": "One MARL algorithm specifically designed to address this problem, and which has also been shown to converge in non-stationary environments, is the weighted policy learner (WPL) algorithm (Abdallah & Lesser, 2006), shown in Algorithm 3. WPL is a PG algorithm designed to dampen oscillation and push the agents to converge more quickly. This is done by scaling the gradient of the expected return for an action $a$ according the probability of taking that action $\\pi ( a )$ (if the gradient is positive) or $1 - { \\overset { - } { \\pi } } ( a )$ (if the gradient is negative). Intuitively, this has the effect of slowing down the learning rate when the policy is moving away from a Nash equilibrium strategy and increasing it when it approaches one. The full WPL algorithm is shown in Algorithm 3. It is assumed that the historical average return $\\hat { \\mathcal { R } } _ { i }$ for each action $a _ { i }$ is initialized to 0 before the start of training. The function simplex-projection projects the updated policy values to make it a valid probability distribution. The projection is defined as: $c l i p ( \\pi ) \\dot { / } \\sum ( c l i p ( \\pi ) )$ , where $c l i p ( x ) = \\operatorname* { m a x } ( 0 , m i n ( 1 , x ) )$ . The states $S$ in the trace are not used by the WPL algorithm. ",
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+ "text": "input : A trace $\\overline { { T = ( S , A , R , r _ { f i n a l } ) } }$ $n$ the maximum depth; $\\hat { \\mathcal { R } }$ , the historical average returns (initialized to 0 at the start of training); $\\gamma$ the discount factor ; and $\\lambda _ { \\pi }$ the policy learning rate output: An updated router policy $\\pi$ 1 for each action $a _ { i } \\in A$ do 2 $\\begin{array} { r } { \\mathcal { R } _ { i } r _ { f i n a l } + \\sum _ { j = i } ^ { n } \\gamma ^ { j - i } r _ { j } } \\end{array}$ 4 Update the average return: 5 $\\hat { \\mathcal R } _ { i } \\gets ( 1 - \\lambda _ { \\pi } ) \\hat { \\mathcal R } _ { i } + \\lambda _ { \\pi } \\mathcal R _ { i }$ 6 Compute the gradient: 7 $\\Delta ( a _ { i } ) \\gets \\mathcal { R } _ { i } - \\hat { \\mathcal { R } } _ { i }$ 8 Update the policy: 9 if $\\Delta ( a _ { i } ) < 0$ then 10 $\\begin{array} { r l } { | } & { { } \\dot { \\Delta } ( \\dot { a } _ { i } ) \\gets \\Delta ( a _ { i } ) ( 1 - \\pi ( a _ { i } ) ) } \\end{array}$ 11 else 12 $\\begin{array} { r l } & { \\dot { \\mathbf { \\bigcup } } \\Delta ( a _ { i } ) \\gets \\Delta ( a _ { i } ) ( \\pi ( a _ { i } ) ) } \\\\ & { \\pi \\gets \\mathrm { s i m p l e x – p r o j e c t i o n } ( \\pi + \\lambda _ { \\pi } \\Delta ) } \\end{array}$ 13 ",
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+ "type": "text",
+ "text": "Details, including convergence proofs and more examples giving the intuition behind the algorithm, can be found in (Abdallah & Lesser, 2006). A longer explanation of the algorithm can be found in Section 7.4 in the appendix. The WPL-Update algorithm is defined only for the tabular setting. It is future work to adapt it to work with function approximators. ",
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+ "text": "As we have described it, the training of the router and function blocks is performed independently after computing the loss. We have also experimented with adding the gradients from the router choices $\\Delta ( a _ { i } )$ to those for the function blocks which produce their input. We found no advantage but leave a more thorough investigation for future work. ",
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+ "text": "4 QUANTITATIVE RESULTS ",
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+ "text": "We experiment with three datasets: multi-task versions of MNIST (MNIST-MTL) (Lecun et al., 1998), Mini-Imagenet (MIN-MTL) (Vinyals et al., 2016) as introduced by (Ravi & Larochelle, 2017), and CIFAR-100 (CIFAR-MTL) (Krizhevsky, 2009) where we treat the 20 superclasses as tasks. In the binary MNIST-MTL dataset, the task is to differentiate instances of a given class $c$ from non-instances. We create 10 tasks and for each we use 1k instances of the positive class $c$ and 1k each of the remaining 9 negative classes for a total of 10k instances per task during training, which we then test on 200 samples per task (2k samples in total). MIN-MTL is a smaller version of ImageNet (Deng et al., 2009) which is easier to train in reasonable time periods. For mini-ImageNet we randomly choose 50 labels and create tasks from 10 disjoint random subsets of 5 labels each chosen from these. Each label has 800 training instances and 50 testing instances – so 4k training and 250 testing instances per task. For all 10 tasks we have a total of 40k training instances. Finally, ",
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+ "text": "CIFAR-100 has coarse and fine labels for its instances. We follow existing work (Krizhevsky, 2009) creating one task for each of the 20 coarse labels and include 500 instances for each of the corresponding fine labels. There are 20 tasks with a total of $2 . 5 \\mathrm { k }$ instances per task; $2 . 5 \\mathrm { k }$ for training and 500 for testing. All results are reported on the test set and are averaged over 3 runs. The data are summarized in Table 1. ",
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+ "type": "text",
+ "text": "Each of these datasets has interesting characteristics which challenge the learning in different ways. CIFAR-MTL is a “natural” dataset whose tasks correspond to human categories. MIN-MTL is randomly generated so will have less task coherence. This makes positive transfer more difficult to achieve and negative transfer more of a problem. And MNIST-MTL, while simple, has the difficult property that the same instance can appear with different labels in different tasks, causing interference. For example, in the $^ { 6 6 } 0$ vs other digits” task, “0” appears with a positive label but in the “1 vs other digits” task it appears with a negative label. ",
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+ "text": "Our experiments are conducted on a convnet architecture (SimpleConvNet) which appeared recently in (Ravi & Larochelle, 2017). This model has 4 convolutional layers, each consisting of a 3x3 convolution and 32 filters, followed by batch normalization and a ReLU. The convolutional layers are followed by 3 fully connected layers, with 128 hidden units each. Our routed version of the network routes the 3 fully connected layers and for each routed layer we supply one randomly initialized function block per task in the dataset. When we use neural net approximators for the router agents they are always 2 layer MLPs with a hidden dimension of 64. A state $( v , t , i )$ is encoded for input to the approximator by concatenating $v$ with a 1-hot representation of $t$ (if used). That is, encoding(s) $=$ concat $( v , \\mathrm { o n e . h o t } ( t ) )$ . ",
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+ "img_path": "images/d952a1687a508616e7f4e1fbbe307bc64d0df37e07f1eff81591a492a82a1d79.jpg",
+ "table_caption": [
+ "Table 1: Dataset training and testing splits "
+ ],
+ "table_footnote": [],
+ "table_body": "| Name | Num Agents | Policy Representation | Part of State = (v,t,d) Used |
| routing-all-fc | Num Tasks | Tabular (numlayers X num function blocks) | t,d |
| routing-all-fc non-layered | Num Tasks | tabular (num layers X num function blocks) | t,d |
| soft-routing-all-fc | Num Tasks | Vector(num layers) of appox functions | v,t,d |
| dispatched-routing-all-fc | Num Tasks +1 | Vector (num layers) of appox functions+ dispatcher | v,td |
| single-agent-routing-all-fc | 1 | Vector (num layers)of approx functions) | v,t,d |
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+ "text": "Policy Representation There are two dominant representation variations, as described in 3.1. In the first, the policy is stored as a table. Since the table needs to store values for each of the different layers of the routing network, it is of size num layers $\\times$ num actions. In the second, it is represented either as vector of MLP’s with a hidden layer of dimension 64, one for each layer of the routing network. In this case the input to the MLP is the representation vector $v$ concatenated with a one-hot representation of the task identifier. ",
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+ "text": "Policy Input describes which parts of the state are used in the decision of the routing action. For tabular policies, the task is used to index the agent responsible for handling that task. Each agent then uses the depth as a row index into into the table. For approximation-based policies, there are two variations. For the single agent case the depth is used to index an approximation function which takes as input concat $_ v$ , one-hot $\\mathbf { \\eta } ^ { ( t ) }$ ). For the multi-agent (non-dispatched) case the task label is used to index the agent and then the depth is used to index the corresponding approximation function for that depth, which is given concat(v, one-hot $\\mathbf { \\rho } ( t )$ ) as input. In the dispatched case, the dispatcher is given concat $\\dot { \\boldsymbol { v } }$ , one-hot(t)) and predicts an agent index. That agent uses the depth to find the approximation function for that depth which is then given concat( $\\boldsymbol { v }$ , one-hot $\\mathbf { \\Psi } ( t ) .$ ) as input. ",
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+ "text": "The WPL algorithm is a multi-agent policy gradient algorithm designed to help dampen policy oscillation and encourage convergence. It does this by slowly scaling down the learning rate for an agent after a gradient change in that agents policy. It determines when there has been a gradient change by using the difference between the immediate reward and historical average reward for the action taken. Depending on the sign of the gradient the algorithm is in one of two scenarios. If the gradient is positive then it is scaled by $1 - \\pi ( a _ { i } )$ . Over time if the gradient remains positive it will cause $\\pi ( \\boldsymbol { a } _ { i } )$ to increase and so $1 - \\pi ( a _ { i } )$ will go to 0, slowing the learning. If the gradient is negative then it is scaled by $\\pi ( \\boldsymbol { a } _ { i } )$ . Here again if the gradient remains negative over time it will cause $\\pi ( \\boldsymbol { a } _ { i } )$ to decrease eventually to 0, slowing the learning again. Slowing the learning after gradient changes dampens the policy oscillation and helps drive the policies towards convergence. ",
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