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Register a new userSample Amplification: Increasing Dataset Size even when Learning is Impossible
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Shivam Garg
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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Given data drawn from an unknown distribution, $D$, to what extent is it possible to ``amplify this dataset and output an even larger set of samples that appear to have been drawn from $D$? We formalize this question as follows: an $(n,m)$ $text{amplification procedure}$ takes as input $n$ independent draws from an unknown distribution $D$, and outputs a set of $m > n$ ``samples. An amplification procedure is valid if no algorithm can distinguish the set of $m$ samples produced by the amplifier from a set of $m$ independent draws from $D$, with probability greater than $2/3$. Perhaps surprisingly, in many settings, a valid amplification procedure exists, even when the size of the input dataset, $n$, is significantly less than what would be necessary to learn $D$ to non-trivial accuracy. Specifically we consider two fundamental settings: the case where $D$ is an arbitrary discrete distribution supported on $le k$ elements, and the case where $D$ is a $d$-dimensional Gaussian with unknown mean, and fixed covariance. In the first case, we show that an $left(n, n + Theta(frac{n}{sqrt{k}})right)$ amplifier exists. In particular, given $n=O(sqrt{k})$ samples from $D$, one can output a set of $m=n+1$ datapoints, whose total variation distance from the distribution of $m$ i.i.d. draws from $D$ is a small constant, despite the fact that one would need quadratically more data, $n=Theta(k)$, to learn $D$ up to small constant total variation distance. In the Gaussian case, we show that an $left(n,n+Theta(frac{n}{sqrt{d}} )right)$ amplifier exists, even though learning the distribution to small constant total variation distance requires $Theta(d)$ samples. In both the discrete and Gaussian settings, we show that these results are tight, to constant factors. Beyond these results, we formalize a number of curious directions for future research along this vein.
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Agents trained by reinforcement learning (RL) often fail to generalize beyond the environment they were trained in, even when presented with new scenarios that seem similar to the training environment. We study the query complexity required to train RL agents that generalize to multiple environments. Intuitively, tractable generalization is only possible when the environments are similar or close in some sense. To capture this, we introduce Weak Proximity, a natural structural condition that requires the environments to have highly similar transition and reward functions and share a policy providing optimal value. Despite such shared structure, we prove that tractable generalization is impossible in the worst case. This holds even when each individual environment can be efficiently solved to obtain an optimal linear policy, and when the agent possesses a generative model. Our lower bound applies to the more complex task of representation learning for the purpose of efficient generalization to multiple environments. On the positive side, we introduce Strong Proximity, a strengthened condition which we prove is sufficient for efficient generalization.
This paper provides a statistical analysis of high-dimensional batch Reinforcement Learning (RL) using sparse linear function approximation. When there is a large number of candidate features, our result sheds light on the fact that sparsity-aware methods can make batch RL more sample efficient. We first consider the off-policy policy evaluation problem. To evaluate a new target policy, we analyze a Lasso fitted Q-evaluation method and establish a finite-sample error bound that has no polynomial dependence on the ambient dimension. To reduce the Lasso bias, we further propose a post model-selection estimator that applies fitted Q-evaluation to the features selected via group Lasso. Under an additional signal strength assumption, we derive a sharper instance-dependent error bound that depends on a divergence function measuring the distribution mismatch between the data distribution and occupancy measure of the target policy. Further, we study the Lasso fitted Q-iteration for batch policy optimization and establish a finite-sample error bound depending on the ratio between the number of relevant features and restricted minimal eigenvalue of the datas covariance. In the end, we complement the results with minimax lower bounds for batch-data policy evaluation/optimization that nearly match our upper bounds. The results suggest that having well-conditioned data is crucial for sparse batch policy learning.
Adversarial attacks during the testing phase of neural networks pose a challenge for the deployment of neural networks in security critical settings. These attacks can be performed by adding noise that is imperceptible to humans on top of the original data. By doing so, an attacker can create an adversarial sample, which will cause neural networks to misclassify. In this paper, we seek to understand the theoretical limits of what can be learned by neural networks in the presence of an adversary. We first defined the hypothesis space of a neural network, and showed the relationship between the growth number of the entire neural network and the growth number of each neuron. Combine that with the adversarial Vapnik-Chervonenkis(VC)-dimension of halfspace classifiers, we concluded the adversarial VC-dimension of the neural networks with sign activation functions.
Suppose we have randomized decision trees for an outer function $f$ and an inner function $g$. The natural approach for obtaining a randomized decision tree for the composed function $(fcirc g^n)(x^1,ldots,x^n)=f(g(x^1),ldots,g(x^n))$ involves amplifying the success probability of the decision tree for $g$, so that a union bound can be used to bound the error probability over all the coordinates. The amplification introduces a logarithmic factor cost overhead. We study the question: When is this log factor necessary? We show that when the outer function is parity or majority, the log factor can be necessary, even for models that are more powerful than plain randomized decision trees. Our results are related to, but qualitatively strengthen in various ways, known results about decision trees with noisy inputs.
Numerous recent works show that overparameterization implicitly reduces variance for min-norm interpolators and max-margin classifiers. These findings suggest that ridge regularization has vanishing benefits in high dimensions. We challenge this narrative by showing that, even in the absence of noise, avoiding interpolation through ridge regularization can significantly improve generalization. We prove this phenomenon for the robust risk of both linear regression and classification and hence provide the first theoretical result on robust overfitting.
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