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Gradient-based learning drives robust representations in recurrent neural networks by balancing compression and expansion

Nature Machine Intelligence volume 4, pages 564–573 (2022)Cite this article

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Abstract

Neural networks need the right representations of input data to learn. Here we ask how gradient-based learning shapes a fundamental property of representations in recurrent neural networks (RNNs)—their dimensionality. Through simulations and mathematical analysis, we show how gradient descent can lead RNNs to compress the dimensionality of their representations in a way that matches task demands during training while supporting generalization to unseen examples. This can require an expansion of dimensionality in early timesteps and compression in later ones, and strongly chaotic RNNs appear particularly adept at learning this balance. Beyond helping to elucidate the power of appropriately initialized artificial RNNs, this fact has implications for neurobiology as well. Neural circuits in the brain reveal both high variability associated with chaos and low-dimensional dynamical structures. Taken together, our findings show how simple gradient-based learning rules lead neural networks to solve tasks with robust representations that generalize to new cases.

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Fig. 1: Task and model schematic.
Fig. 2: Dynamical and geometric properties of networks learning to classify high-dimensional inputs.
Fig. 3: Dynamical and geometric properties of networks learning to classify two-dimensional inputs.
Fig. 4: Dynamical and geometric properties of networks learning to classify two-dimensional inputs restricted to two neurons.
Fig. 5: Networks with mean squared error loss and linear units continue to compress dimensionality.
Fig. 6: Variability of gradients compresses neural representations in networks with injected hidden unit noise and regularization.

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All data used in the paper are generated by the code at refs. 61.

Code availability

Code for training the networks and generating the plots can be found in a Code Ocean capsule61.

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Acknowledgements

M.F. was funded by the National Science Foundation Graduate Research Fellowship under Grant DGE-1256082. G.L. is funded by an NSERC Discovery Grant (RGPIN-2018-04821), an FRQNT Young Investigator Startup Program (2019-NC-253251) and an FRQS Research Scholar Award, Junior 1 (LAJGU0401-253188). E.S.-B. acknowledges the support of NSF DMS Grant 1514743. M.F. thanks the Swartz Program in Theoretical Neuroscience at Harvard and S.R. thanks the Swartz Center for Theoretical Neuroscience at the University of Washington for support. We thank M. Stern, D. Chklovskii, A. Weber, N. Steinmetz and L. Mazzucato for their insights and suggestions. M.F. would also like to thank H. Sompolinsky and S. Chung for their mentorship and inspiration.

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  1. Matthew Farrell

    Present address: John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA

Authors and Affiliations

  1. Department of Applied Mathematics, University of Washington, Seattle, WA, USA

    Matthew Farrell & Eric Shea-Brown

  2. Computational Neuroscience Center, University of Washington, Seattle, WA, USA

    Matthew Farrell, Stefano Recanatesi, Timothy Moore & Eric Shea-Brown

  3. Mila, Montreal, Quebec, Canada

    Guillaume Lajoie

  4. Département de mathématiques et de statistique, Université de Montréal Pavillon André-Aisenstadt, Montreal, Quebec, Canada

    Guillaume Lajoie

  5. Allen Institute for Brain Science, Seattle, WA, USA

    Eric Shea-Brown

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Contributions

M.F., S.R. and E.S.-B. conceived the study. M.F. wrote code and ran simulations with some guidance from S.R. The manuscript was primarily written by M.F., with substantial edits and contributions made by S.R., G.L. and E.S.-B. G.L. contributed code for computing Lyapunov exponents and provided additional insight. T.M. ran the simulations for Extended Data Fig. 9 and ran additional verification experiments for intermediate values of β.

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Extended data

Extended Data Fig. 1 Effects of changing the evaluation timestep and number of recurrent units.

Strongly chaotic (red) and edge-of-chaos (cyan) networks are trained to classify high-dimensional inputs. Details are as in Fig. 2e. Shaded regions are as defined in Fig. 2e. First row: network trained with a categorical cross-entropy loss with a learning rate of 1e-4. Second row: network trained with a mean squared error loss with a learning rate of 1e-3. First column: evaluation time is t = 6. Second column: evaluation time is t = 10. Third column: evaluation time is t = 14. Fourth column: Number of hidden neurons is increased to N = 300. Evaluation time is t = 14.

Extended Data Fig. 2 Effects of changing the evaluation timestep, input dimension, and number of recurrent units.

Strongly chaotic (red) and edge-of-chaos (cyan) networks are trained to classify low-dimensional inputs. Dashed lines depict the network before training, and solid is after training. All networks are trained with a categorical cross-entropy loss and a learning rate of 1e-4 (note that this is a factor of 10 less than used in the main text). Shaded regions are as defined in Fig. 2e. Other details are as in Fig. 3e. First row: 2-dimensional inputs. Second row: 4-dimensional inputs. Third row: 10-dimensional inputs. First column: evaluation time is t = 6. Second column: evaluation time is t = 10. Third column: evaluation time is t = 14. Fourth column: Number of hidden neurons is increased to N = 300. Evaluation time is t = 14.

Extended Data Fig. 3 Effects of changing the evaluation timestep, input dimension, and number of recurrent units on logistic regression testing accuracy.

Strongly chaotic (red) and edge-of-chaos (cyan) networks are trained to classify low-dimensional inputs. Dashed lines depict the network before training, and solid is after training. Details are as in Extended Data Fig. 2, but this time measuring the logistic regression testing accuracy as defined in the main text.

Extended Data Fig. 4 Effects of changing the evaluation timestep, input dimension, and number of recurrent units with 120 input clusters.

Strongly chaotic (red) and edge-of-chaos (cyan) networks are trained to classify low-dimensional inputs. Dashed lines depict the network before training, and solid is after training. Details are as in Extended Data Fig. 2, but here 120 input clusters are used instead of 60.

Extended Data Fig. 5 Between-class distances are increased while within class distances are diminished by the network dynamics.

Mean pairwise distance between points belonging to the same class (dashed lines), mean pairwise distance between points belonging to different classes (dotted lines), and the ratio of the first to the second (blue lines and axes), for the representations of trained networks over time t. Details are as in Fig. 2e in the main text. Shaded regions are as defined in Fig. 2e. a. Edge-of-chaos network as defined in the main text. b. Strongly chaotic network as defined in the main text.

Extended Data Fig. 6 Dependence of dimensionality on the learning rate.

Here we reproduce the results of Figs. 2e and 3e of the main text, using a different learning rate. Red lines correspond to strongly chaotic and cyan lines to edge-of-chaos networks. Dashed and solid lines depict before and after training, respectively. Shaded regions are as defined in Fig. 2e. Top row: high-dimensional inputs as in Fig. 2e. Bottom row: low-dimensional inputs as in Fig. 3e. Left column: learning rate of 1e-4. Right column: learning rate of 1e-3, as in the main text.

Extended Data Fig. 7 Dimensionality increases with number of class labels, but not number of clusters.

Effective dimensionalities (EDs) of the trained network responses to inputs embedded in an N-dimensional space, measured at the evaluation time teval = 10. Error bars denote two standard deviations of three initializations of task and networks (in all panels they are too small to see). Details are similar to Fig. 2. a. Edge-of-chaos networks. Blue: ED of the inputs. Green: ED of the network representation as a function of the number of input clusters. Dimensionality remains flat and small. b. Edge-of-chaos networks. Green: ED of the network representation as a function of the number of class labels. Black: Effective dimensionality of points distributed uniformly at random in an N-dimensional ball. The number of points drawn is determined by the number of class labels. This is to roughly measure what the ED of the network would be if it formed a fixed point for every class label, and distributed these fixed points randomly in space. c. Strongly chaotic networks. Legend as in a. d. Strongly chaotic networks. Legend as in b.

Extended Data Fig. 8 Example of noise in output weights driving compression of the hidden representation in a linear network with two hidden layer units.

The equation for the network is h = Wx + b with output \(\hat{o}={{{{\boldsymbol{r}}}}}^{T}{{{\boldsymbol{h}}}}\). The input weights (red) are initialized to the 2 × 2 identity matrix, and bias is initialized as (1, 0). The inputs are placed on a grid from x = − 1 to x = 2 and from y = − 3 to y = 3 (not shown). Network output \(\hat{o}\) is trained to minimize the squared error loss \(0.5{(\hat{o}-1)}^{2}\). Input samples are chosen randomly, and input weights are updated via stochastic gradient descent with batch size 1. a. Top: Diagram of network where input weights are trained and output weights are fixed. Bottom: Diagram of network where input weights are trained and output weights are drawn from a normal distributed with mean (1, 0) and covariance 0.05I at every update step. In the figure, η represents additive white noise. Middle: hidden unit responses (blue circles) to the inputs before training (iteration 0). Black dot denotes the output weight vector, and the blue line is the affine subspace of points that r maps to 1. b. Evolution of the hidden layer response to inputs (representation) as input weights are trained. Top: Representation of the network where output weights are fixed. The iteration number denotes the number of training samples that have been used to update the weights. Activations compress to the space orthogonal to r, shifted by (1, 0). Bottom: Representation of the network where output weights are randomly drawn at every input sample presentation. Activations compress to a compact, localized space. The direction of compression is both along and orthogonal to r.

Extended Data Fig. 9 Representations of RNNs trained on the MNIST digit recognition dataset.

a. Effective dimensionality (ED) of RNNs trained on the MNIST digit recognition dataset. The ED of the network’s responses to test inputs is plotted. After training, dimensionality compresses down to a value by t = 10 that roughly matches the number of class labels (10). This compression is similar to that seen in Fig. 2 of the main text. Details are as in Fig. 2e of the main text. Shaded regions are as defined in Fig. 2e. b. Projection onto the top three principal components of MNIST test data. Colours indicate true class label (i.e., digit identity). c. Projection onto the top three principal components of the edge-of-chaos recurrent network’s responses to the inputs in b after training, at the evaluation time t = 10. Colours indicate true class label as in b. The network forms a localized cluster for each digit.

Extended Data Fig. 10 Effects of changing the initial coupling strength β.

Details are as in Fig. 3e, except that here we vary the coupling parameter β, whose value is indicated by the colourbar to the right. Shaded regions are as defined in Fig. 2e. First column: ED of networks before training. Second column: ED of networks after training with a learning rate of 1e-3. Third column: ED of networks after training with a learning rate of 1e-4.

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Farrell, M., Recanatesi, S., Moore, T. et al. Gradient-based learning drives robust representations in recurrent neural networks by balancing compression and expansion. Nat Mach Intell 4, 564–573 (2022). https://doi.org/10.1038/s42256-022-00498-0

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