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We study the power of learning via mini-batch stochastic gradient descent (SGD) on the population loss, and batch Gradient Descent (GD) on the empirical loss, of a differentiable model or neural network, and ask what learning problems can be learnt using these paradigms. We show that SGD and GD can always simulate learning with statistical queries (SQ), but their ability to go beyond that depends on the precision $rho$ of the gradient calculations relative to the minibatch size $b$ (for SGD) and sample size $m$ (for GD). With fine enough precision relative to minibatch size, namely when $b rho$ is small enough, SGD can go beyond SQ learning and simulate any sample-based learning algorithm and thus its learning power is equivalent to that of PAC learning; this extends prior work that achieved this result for $b=1$. Similarly, with fine enough precision relative to the sample size $m$, GD can also simulate any sample-based learning algorithm based on $m$ samples. In particular, with polynomially many bits of precision (i.e. when $rho$ is exponentially small), SGD and GD can both simulate PAC learning regardless of the mini-batch size. On the other hand, when $b rho^2$ is large enough, the power of SGD is equivalent to that of SQ learning.
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 algo
The study of strategic or adversarial manipulation of testing data to fool a classifier has attracted much recent attention. Most previous works have focused on two extreme situations where any testing data point either is completely adversarial or a
Bayesian structure learning allows inferring Bayesian network structure from data while reasoning about the epistemic uncertainty -- a key element towards enabling active causal discovery and designing interventions in real world systems. In this wor
We introduce a new and rigorously-formulated PAC-Bayes few-shot meta-learning algorithm that implicitly learns a prior distribution of the model of interest. Our proposed method extends the PAC-Bayes framework from a single task setting to the few-sh
PAC-learning usually aims to compute a small subset ($varepsilon$-sample/net) from $n$ items, that provably approximates a given loss function for every query (model, classifier, hypothesis) from a given set of queries, up to an additive error $varep