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Register a new userMinimax Lower Bounds for Transfer Learning with Linear and One-hidden Layer Neural Networks
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Seyed Mohammadreza Mousavi Kalan
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Transfer learning has emerged as a powerful technique for improving the performance of machine learning models on new domains where labeled training data may be scarce. In this approach a model trained for a source task, where plenty of labeled training data is available, is used as a starting point for training a model on a related target task with only few labeled training data. Despite recent empirical success of transfer learning approaches, the benefits and fundamental limits of transfer learning are poorly understood. In this paper we develop a statistical minimax framework to characterize the fundamental limits of transfer learning in the context of regression with linear and one-hidden layer neural network models. Specifically, we derive a lower-bound for the target generalization error achievable by any algorithm as a function of the number of labeled source and target data as well as appropriate notions of similarity between the source and target tasks. Our lower bound provides new insights into the benefits and limitations of transfer learning. We further corroborate our theoretical finding with various experiments.
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We study the problem of PAC learning one-hidden-layer ReLU networks with $k$ hidden units on $mathbb{R}^d$ under Gaussian marginals in the presence of additive label noise. For the case of positive coefficients, we give the first polynomial-time algorithm for this learning problem for $k$ up to $tilde{O}(sqrt{log d})$. Previously, no polynomial time algorithm was known, even for $k=3$. This answers an open question posed by~cite{Kliv17}. Importantly, our algorithm does not require any assumptions about the rank of the weight matrix and its complexity is independent of its condition number. On the negative side, for the more general task of PAC learning one-hidden-layer ReLU networks with arbitrary real coefficients, we prove a Statistical Query lower bound of $d^{Omega(k)}$. Thus, we provide a separation between the two classes in terms of efficient learnability. Our upper and lower bounds are general, extending to broader families of activation functions.
We study the complexity of training neural network models with one hidden nonlinear activation layer and an output weighted sum layer. We analyze Gradient Descent applied to learning a bounded target function on $n$ real-valued inputs. We give an agnostic learning guarantee for GD: starting from a randomly initialized network, it converges in mean squared loss to the minimum error (in $2$-norm) of the best approximation of the target function using a polynomial of degree at most $k$. Moreover, for any $k$, the size of the network and number of iterations needed are both bounded by $n^{O(k)}log(1/epsilon)$. In particular, this applies to training networks of unbiased sigmoids and ReLUs. We also rigorously explain the empirical finding that gradient descent discovers lower frequency Fourier components before higher frequency components. We complement this result with nearly matching lower bounds in the Statistical Query model. GD fits well in the SQ framework since each training step is determined by an expectation over the input distribution. We show that any SQ algorithm that achieves significant improvement over a constant function with queries of tolerance some inverse polynomial in the input dimensionality $n$ must use $n^{Omega(k)}$ queries even when the target functions are restricted to a set of $n^{O(k)}$ degree-$k$ polynomials, and the input distribution is uniform over the unit sphere; for this class the information-theoretic lower bound is only $Theta(k log n)$. Our approach for both parts is based on spherical harmonics. We view gradient descent as an operator on the space of functions, and study its dynamics. An essential tool is the Funk-Hecke theorem, which explains the eigenfunctions of this operator in the case of the mean squared loss.
We propose a new algorithm to learn a one-hidden-layer convolutional neural network where both the convolutional weights and the outputs weights are parameters to be learned. Our algorithm works for a general class of (potentially overlapping) patches, including commonly used structures for computer vision tasks. Our algorithm draws ideas from (1) isotonic regression for learning neural networks and (2) landscape analysis of non-convex matrix factorization problems. We believe these findings may inspire further development in designing provable algorithms for learning neural networks and other complex models.
Dictionary learning is the problem of estimating the collection of atomic elements that provide a sparse representation of measured/collected signals or data. This paper finds fundamental limits on the sample complexity of estimating dictionaries for tensor data by proving a lower bound on the minimax risk. This lower bound depends on the dimensions of the tensor and parameters of the generative model. The focus of this paper is on second-order tensor data, with the underlying dictionaries constructed by taking the Kronecker product of two smaller dictionaries and the observed data generated by sparse linear combinations of dictionary atoms observed through white Gaussian noise. In this regard, the paper provides a general lower bound on the minimax risk and also adapts the proof techniques for equivalent results using sparse and Gaussian coefficient models. The reported results suggest that the sample complexity of dictionary learning for tensor data can be significantly lower than that for unstructured data.
Given a large data matrix $Ainmathbb{R}^{ntimes n}$, we consider the problem of determining whether its entries are i.i.d. with some known marginal distribution $A_{ij}sim P_0$, or instead $A$ contains a principal submatrix $A_{{sf Q},{sf Q}}$ whose entries have marginal distribution $A_{ij}sim P_1 eq P_0$. As a special case, the hidden (or planted) clique problem requires to find a planted clique in an otherwise uniformly random graph. Assuming unbounded computational resources, this hypothesis testing problem is statistically solvable provided $|{sf Q}|ge C log n$ for a suitable constant $C$. However, despite substantial effort, no polynomial time algorithm is known that succeeds with high probability when $|{sf Q}| = o(sqrt{n})$. Recently Meka and Wigderson cite{meka2013association}, proposed a method to establish lower bounds within the Sum of Squares (SOS) semidefinite hierarchy. Here we consider the degree-$4$ SOS relaxation, and study the construction of cite{meka2013association} to prove that SOS fails unless $kge C, n^{1/3}/log n$. An argument presented by Barak implies that this lower bound cannot be substantially improved unless the witness construction is changed in the proof. Our proof uses the moments method to bound the spectrum of a certain random association scheme, i.e. a symmetric random matrix whose rows and columns are indexed by the edges of an Erdos-Renyi random graph.
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