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Title: Deep Ensembles: A Loss Landscape Perspective

Paper content:
Deep ensembles have been empirically shown to be a promising approach for improving accuracy, uncertainty  and out-of-distribution robustness of deep learning models.
While deep ensembles were theoretically motivated by the bootstrap, non-bootstrap ensembles trained with just random initialization also perform well in practice, which suggests that there could be other explanations for why deep ensembles work well.
Bayesian neural networks, which learn distributions over the parameters of the network, are theoretically well-motivated by Bayesian principles, but do not perform as well as deep ensembles in practice, particularly under dataset shift.
One possible explanation for this gap between theory and practice is that popular scalable approximate Bayesian methods tend to focus on a single mode, whereas deep ensembles tend to explore diverse modes in function space.
We investigate this hypothesis by building on recent work on understanding the loss landscape of neural networks and adding our own exploration to measure the similarity of functions in the space of predictions.
Our results show that random initializations explore entirely different modes, while functions along an optimization trajectory or sampled from the subspace thereof cluster within a single mode predictions-wise, while often deviating significantly in the weight space.
We demonstrate that while low-loss connectors between modes exist, they are not connected in the space of predictions.
Developing the concept of the diversity--accuracy plane, we show that the decorrelation power of random initializations is unmatched by popular subspace sampling methods.
Consider a typical classification problem, where x n ∈ R D denotes the D-dimensional features and y n ∈ [1, . . . , K] denotes the class label.
Assume we have a parametric model p(y|x, θ) for the conditional distribution where θ denotes weights and biases of a neural network, and p(θ) is a prior distribution over parameters.
The Bayesian posterior over parameters is given by
p(y n |x n , θ).
(
Computing the exact posterior distribution over θ is computationally expensive (if not impossible) when p(y n |x n , θ) is a deep neural network.
A variety of approximations have been developed for Bayesian neural networks, including Laplace approximation (MacKay, 1992) , Markov chain Monte Carlo methods (Neal, 1996; Welling & Teh, 2011; Springenberg et al., 2016) , variational Bayesian methods (Graves, 2011; Blundell et al., 2015; Louizos & Welling, 2017; Wen et al., 2018) and Monte-Carlo dropout (Gal & Ghahramani, 2016; Srivastava et al., 2014) .
While computing the posterior is challenging, it is usually easy to perform maximum-a-posteriori (MAP) estimation, which corresponds to a mode of the posterior.
The MAP solution can be written as the minimizer of the following loss (negative log likelihood + negative log prior):
log p(y n |x n , θ).
The MAP solution is computationally efficient, but only gives a point estimate and not a distribution over parameters.
Deep ensembles, proposed by Lakshminarayanan et al. (2017) , train an ensemble of neural networks by initializing at M different values and repeating the minimization multiple times which could lead to M different solutions, if the loss is non-convex.
(Lakshminarayanan et al. (2017) found adversarial training provides additional benefits in some of their experiments, but we will ignore adversarial training and focus only on ensembles with random initialization in this paper.
)
Given finite training data, many parameter values could equally well explain the observations, and capturing these diverse solutions is crucial for quantifying epistemic uncertainty (Kendall & Gal, 2017) .
Bayesian neural networks learn a distribution over weights, and a good posterior approximation should be able to learn multi-modal posterior distributions in theory.
Deep ensembles were inspired by the bootstrap (Breiman, 1996) , which has nice theoretical properties.
However, it has been empirically observed by Lakshminarayanan et al. (2017) ; Lee et al. (2015) that training individual networks with just random initialization is sufficient in practice and using the bootstrap even hurts performance in some cases (e.g. for small ensemble sizes).
Furthermore, Ovadia et al. (2019) and Gustafsson et al. (2019) independently benchmarked existing methods for uncertainty quantification on a variety of datasets and architectures, and observed that ensembles tend to outperform approximate Bayesian neural networks in terms of both accuracy and uncertainty, particularly under dataset shift.
) on train and validation data.
These empirical observations raise an important question: Why do ensembles trained with just random initialization work so well in practice?
One possible hypothesis is that ensembles tend to sample from different modes 1 in function space, whereas variational Bayesian methods (which minimize
)) might fail to explore multiple modes even though they are effective at capturing uncertainty within a single mode.
See Figure 1 for a cartoon illustration.
Note that while the MAP solution is a local minima for the training loss by definition, it may not necessarily be a local minima for the validation loss.
Recent work on understanding loss landscapes (Fort & Jastrzebski, 2019; Draxler et al., 2018; allows us to investigate this hypothesis.
Note that prior work on loss landscapes has focused on mode-connectivity and low-loss tunnels, but has not explicitly focused on how diverse the functions from different modes are, beyond an initial exploration in Fort & Jastrzebski (2019) .
Our findings show that:
• The functions sampled along a single training trajectory or subspace thereof (e.g. diagonal Gaussian, low-rank Gaussian and Dropout subspaces) tend to be very similar in predictions (while potential far away in the weight space), whereas functions sampled from different randomly initialized trajectories tend to be very diverse.
• Solution modes are connected in the loss landscape but they are distinct in the space of predictions.
Low-loss tunnels create functions with near-identical low values of loss along the path, however these functions tend to be very different in function space, changing significantly in the middle of the tunnel.
Our results show that trajectories of randomly initialized neural networks explore different modes in function space, which explains why deep ensembles with random initializations help.
They are essentially orthogonal to each other in the space of weights and very diverse in terms of their predictions.
While these modes can be connected via optimized low-loss paths between them, we demonstrate that they correspond to distinct functions in terms of their predictions.
Therefore the connectivity in the loss landscape does not imply connectivity in the space of functions.
Subspace sampling methods such as weight averaging, Monte Carlo dropout, and various versions of local Gaussian approximations, sample functions that might lie relatively far from the starting point in the weight space, however, they remain in the vicinity of their starting point in terms of predictions, giving rise to an insufficiently diverse set of functions.
Using the concept of the diversityaccuracy plane, we demonstrate empirically that these subspace sampling methods never reach the combination of diversity and accuracy that independently trained models do, limiting their usefulness for ensembling.
A VISUALIZING THE LOSS LANDSCAPE ALONG ORIGINAL DIRECTIONS AND WA DIRECTIONS Figure S1 shows the loss landscape (train as well as the validation set) and the effect of WA.
Figure S1 : Loss landscape versus generalization: weights are typically initialized close to 0 and increase radially through the course of training.
Top row: we pick two optima from different trajectories as the axes, and plot loss surface.
Looking at x and y axes, we observe that while a wide range of radii achieve low loss on training set, the range of optimal radius values is narrower on validation set.
Bottom row: we average weights within each trajectory using WA and use them as axes.
A wider range of radius values generalize better along the WA directions, which confirms the findings of .

assistant: text: We study deep ensembles through the lens of loss landscape and the space of predictions, demonstrating that the decorrelation power of random initializations is unmatched by subspace sampling that only explores a single mode.