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Register a new userThe Bias-Expressivity Trade-off
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Learning algorithms need bias to generalize and perform better than random guessing. We examine the flexibility (expressivity) of biased algorithms. An expressive algorithm can adapt to changing training data, altering its outcome based on changes in its input. We measure expressivity by using an information-theoretic notion of entropy on algorithm outcome distributions, demonstrating a trade-off between bias and expressivity. To the degree an algorithm is biased is the degree to which it can outperform uniform random sampling, but is also the degree to which is becomes inflexible. We derive bounds relating bias to expressivity, proving the necessary trade-offs inherent in trying to create strongly performing yet flexible algorithms.
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Breakthroughs in machine learning are rapidly changing science and society, yet our fundamental understanding of this technology has lagged far behind. Indeed, one of the central tenets of the field, the bias-variance trade-off, appears to be at odds with the observed behavior of methods used in the modern machine learning practice. The bias-variance trade-off implies that a model should balance under-fitting and over-fitting: rich enough to express underlying structure in data, simple enough to avoid fitting spurious patterns. However, in the modern practice, very rich models such as neural networks are trained to exactly fit (i.e., interpolate) the data. Classically, such models would be considered over-fit, and yet they often obtain high accuracy on test data. This apparent contradiction has raised questions about the mathematical foundations of machine learning and their relevance to practitioners. In this paper, we reconcile the classical understanding and the modern practice within a unified performance curve. This double descent curve subsumes the textbook U-shaped bias-variance trade-off curve by showing how increasing model capacity beyond the point of interpolation results in improved performance. We provide evidence for the existence and ubiquity of double descent for a wide spectrum of models and datasets, and we posit a mechanism for its emergence. This connection between the performance and the structure of machine learning models delineates the limits of classical analyses, and has implications for both the theory and practice of machine learning.
Modern Monte Carlo-type approaches to dynamic decision problems are reformulated as empirical loss minimization, allowing direct applications of classical results from statistical machine learning. These computational methods are then analyzed in this framework to demonstrate their effectiveness as well as their susceptibility to generalization error. Standard uses of classical results prove potential overlearning, thus bias-variance trade-off, by connecting over-trained networks to anticipating controls. On the other hand, non-asymptotic estimates based on Rademacher complexity show the convergence of these algorithms for sufficiently large training sets. A numerically studied stylized example illustrates these possibilities, including the importance of problem dimension in the degree of overlearning, and the effectiveness of this approach.
We empirically investigate the best trade-off between sparse and uniformly-weighted multiple kernel learning (MKL) using the elastic-net regularization on real and simulated datasets. We find that the best trade-off parameter depends not only on the sparsity of the true kernel-weight spectrum but also on the linear dependence among kernels and the number of samples.
The overall performance or expected excess risk of an iterative machine learning algorithm can be decomposed into training error and generalization error. While the former is controlled by its convergence analysis, the latter can be tightly handled by algorithmic stability. The machine learning community has a rich history investigating convergence and stability separately. However, the question about the trade-off between these two quantities remains open. In this paper, we show that for any iterative algorithm at any iteration, the overall performance is lower bounded by the minimax statistical error over an appropriately chosen loss function class. This implies an important trade-off between convergence and stability of the algorithm -- a faster converging algorithm has to be less stable, and vice versa. As a direct consequence of this fundamental tradeoff, new convergence lower bounds can be derived for classes of algorithms constrained with different stability bounds. In particular, when the loss function is convex (or strongly convex) and smooth, we discuss the stability upper bounds of gradient descent (GD) and stochastic gradient descent and their variants with decreasing step sizes. For Nesterovs accelerated gradient descent (NAG) and heavy ball method (HB), we provide stability upper bounds for the quadratic loss function. Applying existing stability upper bounds for the gradient methods in our trade-off framework, we obtain lower bounds matching the well-established convergence upper bounds up to constants for these algorithms and conjecture similar lower bounds for NAG and HB. Finally, we numerically demonstrate the tightness of our stability bounds in terms of exponents in the rate and also illustrate via a simulated logistic regression problem that our stability bounds reflect the generalization errors better than the simple uniform convergence bounds for GD and NAG.
The classical bias-variance trade-off predicts that bias decreases and variance increase with model complexity, leading to a U-shaped risk curve. Recent work calls this into question for neural networks and other over-parameterized models, for which it is often observed that larger models generalize better. We provide a simple explanation for this by measuring the bias and variance of neural networks: while the bias is monotonically decreasing as in the classical theory, the variance is unimodal or bell-shaped: it increases then decreases with the width of the network. We vary the network architecture, loss function, and choice of dataset and confirm that variance unimodality occurs robustly for all models we considered. The risk curve is the sum of the bias and variance curves and displays different qualitative shapes depending on the relative scale of bias and variance, with the double descent curve observed in recent literature as a special case. We corroborate these empirical results with a theoretical analysis of two-layer linear networks with random first layer. Finally, evaluation on out-of-distribution data shows that most of the drop in accuracy comes from increased bias while variance increases by a relatively small amount. Moreover, we find that deeper models decrease bias and increase variance for both in-distribution and out-of-distribution data.
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