We study the problem of agnostic learning under the Gaussian distribution. We develop a method for finding hard families of examples for a wide class of problems by using LP duality. For Boolean-valued concept classes, we show that the $L^1$-regression algorithm is essentially best possible, and therefore that the computational difficulty of agnostically learning a concept class is closely related to the polynomial degree required to approximate any function from the class in $L^1$-norm. Using this characterization along with additional analytic tools, we obtain optimal SQ lower bounds for agnostically learning linear threshold functions and the first non-trivial SQ lower bounds for polynomial threshold functions and intersections of halfspaces. We also develop an analogous theory for agnostically learning real-valued functions, and as an application prove near-optimal SQ lower bounds for agnostically learning ReLUs and sigmoids.
We study the problem of agnostically learning halfspaces under the Gaussian distribution. Our main result is the {\em first proper} learning algorithm for this problem whose sample complexity and computational complexity qualitatively match those of the best known improper agnostic learner. Building on this result, we also obtain the first proper polynomial-time approximation scheme (PTAS) for agnostically learning homogeneous halfspaces. Our techniques naturally extend to agnostically learning linear models with respect to other non-linear activations, yielding in particular the first proper …
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of California, San Diego（加利福尼亚大学圣迭戈分校）
We study the fundamental problems of agnostically learning halfspaces and ReLUs under Gaussian marginals. In the former problem, given labeled examples $(\mathbf{x}, y)$ from an unknown distribution on $\mathbb{R}^d \times \{ \pm 1\}$, whose marginal distribution on $\mathbf{x}$ is the standard Gaussian and the labels $y$ can be arbitrary, the goal is to output a hypothesis with 0-1 loss $\mathrm{OPT}+ε$, where $\mathrm{OPT}$ is the 0-1 loss of the best-fitting halfspace. In the latter problem, given labeled examples $(\mathbf{x}, y)$ from an unknown distribution on $\mathbb{R}^d \times \mathb…
Model-agnostic meta-learning (MAML) formulates meta-learning as a bilevel optimization problem, where the inner level solves each subtask based on a shared prior, while the outer level searches for the optimal shared prior by optimizing its aggregated performance over all the subtasks. Despite its empirical success, MAML remains less understood in theory, especially in terms of its global optimality, due to the nonconvexity of the meta-objective (the outer-level objective). To bridge such a gap between theory and practice, we characterize the optimality gap of the stationary points attained by…
We study the task of agnostically learning halfspaces under the Gaussian distribution. Specifically, given labeled examples $(\mathbf{x},y)$ from an unknown distribution on $\mathbb{R}^n \times \{ \pm 1\}$, whose marginal distribution on $\mathbf{x}$ is the standard Gaussian and the labels $y$ can be arbitrary, the goal is to output a hypothesis with 0-1 loss $\mathrm{OPT}+ε$, where $\mathrm{OPT}$ is the 0-1 loss of the best-fitting halfspace. We prove a near-optimal computational hardness result for this task, under the widely believed sub-exponential time hardness of the Learning with Errors…
The performance of state-of-the-art offline RL methods varies widely over the spectrum of dataset qualities, ranging from far-from-optimal random data to close-to-optimal expert demonstrations. We re-implement these methods to test their reproducibility, and show that when a given method outperforms the others on one end of the spectrum, it never does on the other end. This prevents us from naming a victor across the board. We attribute the asymmetry to the amount of inductive bias injected into the agent to entice it to posit that the behavior underlying the offline dataset is optimal for the…
We consider the well-studied problem of learning intersections of halfspaces under the Gaussian distribution in the challenging \emph{agnostic learning} model. Recent work of Diakonikolas et al. (2021) shows that any Statistical Query (SQ) algorithm for agnostically learning the class of intersections of $k$ halfspaces over $\mathbb{R}^n$ to constant excess error either must make queries of tolerance at most $n^{-\tildeΩ(\sqrt{\log k})}$ or must make $2^{n^{Ω(1)}}$ queries. We strengthen this result by improving the tolerance requirement to $n^{-\tildeΩ(\log k)}$. This lower bound is essential…
One notable weakness of current machine learning algorithms is the poor ability of models to solve new problems without forgetting previously acquired knowledge. The Continual Learning paradigm has emerged as a protocol to systematically investigate settings where the model sequentially observes samples generated by a series of tasks. In this work, we take a task-agnostic view of continual learning and develop a hierarchical information-theoretic optimality principle that facilitates a trade-off between learning and forgetting. We derive this principle from a Bayesian perspective and show its …
University of Oxford（牛津大学）；The Alan Turing Institute（艾伦·图灵研究所）；University of Surrey（萨里大学）
Submodular extensions of an energy function can be used to efficiently compute approximate marginals via variational inference. The accuracy of the marginals depends crucially on the quality of the submodular extension. To identify the best possible extension, we show an equivalence between the submodular extensions of the energy and the objective functions of linear programming (LP) relaxations for the corresponding MAP estimation problem. This allows us to (i) establish the worst-case optimality of the submodular extension for Potts model used in the literature; (ii) identify the worst-case …
Beta coefficients for linear regression models represent the ideal form of an interpretable feature effect. However, for non-linear models and especially generalized linear models, the estimated coefficients cannot be interpreted as a direct feature effect on the predicted outcome. Hence, marginal effects are typically used as approximations for feature effects, either in the shape of derivatives of the prediction function or forward differences in prediction due to a change in a feature value. While marginal effects are commonly used in many scientific fields, they have not yet been adopted a…
Microsoft Research NYC（微软研究院纽约中心）；University of Texas at Austin（得克萨斯大学奥斯汀分校）；Institute for Advanced Study（高等研究院）
We give the first statistical-query lower bounds for agnostically learning any non-polynomial activation with respect to Gaussian marginals (e.g., ReLU, sigmoid, sign). For the specific problem of ReLU regression (equivalently, agnostically learning a ReLU), we show that any statistical-query algorithm with tolerance $n^{-(1/ε)^b}$ must use at least $2^{n^c} ε$ queries for some constant $b, c > 0$, where $n$ is the dimension and $ε$ is the accuracy parameter. Our results rule out general (as opposed to correlational) SQ learning algorithms, which is unusual for real-valued learning problems. O…
We show hardness of improperly learning halfspaces in the agnostic model, both in the distribution-independent as well as the distribution-specific setting, based on the assumption that worst-case lattice problems, such as GapSVP or SIVP, are hard. In particular, we show that under this assumption there is no efficient algorithm that outputs any binary hypothesis, not necessarily a halfspace, achieving misclassfication error better than $\frac 1 2 - γ$ even if the optimal misclassification error is as small is as small as $δ$. Here, $γ$ can be smaller than the inverse of any polynomial in the …
Preconditioned Gradient Descent for Overparameterized Nonconvex Burer--Monteiro Factorization with Global Optimality Certification
Electrical and Computer Engineering University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校电气与计算机工程学院）；Industrial and Operations Engineering University of Michigan（密歇根大学工业与运营工程学院）
We study the problem of agnostically learning homogeneous halfspaces in the distribution-specific PAC model. For a broad family of structured distributions, including log-concave distributions, we show that non-convex SGD efficiently converges to a solution with misclassification error $O(\opt)+\eps$, where $\opt$ is the misclassification error of the best-fitting halfspace. In sharp contrast, we show that optimizing any convex surrogate inherently leads to misclassification error of $ω(\opt)$, even under Gaussian marginals.
We consider the problem of learning the best-fitting single neuron as measured by the expected square loss $\mathbb{E}_{(x,y)\sim \mathcal{D}}[(σ(w^\top x)-y)^2]$ over some unknown joint distribution $\mathcal{D}$ by using gradient descent to minimize the empirical risk induced by a set of i.i.d. samples $S\sim \mathcal{D}^n$. The activation function $σ$ is an arbitrary Lipschitz and non-decreasing function, making the optimization problem nonconvex and nonsmooth in general, and covers typical neural network activation functions and inverse link functions in the generalized linear model settin…
Ulm University（乌尔姆大学）；Institute of Neural Information Processing（神经信息处理研究所）
One weakness of machine learning algorithms is the poor ability of models to solve new problems without forgetting previously acquired knowledge. The Continual Learning (CL) paradigm has emerged as a protocol to systematically investigate settings where the model sequentially observes samples generated by a series of tasks. In this work, we take a task-agnostic view of continual learning and develop a hierarchical information-theoretic optimality principle that facilitates a trade-off between learning and forgetting. We discuss this principle from a Bayesian perspective and show its connection…
We study the problem of approximating and learning coverage functions. A function $c: 2^{[n]} \rightarrow \mathbf{R}^{+}$ is a coverage function, if there exists a universe $U$ with non-negative weights $w(u)$ for each $u \in U$ and subsets $A_1, A_2, \ldots, A_n$ of $U$ such that $c(S) = \sum_{u \in \cup_{i \in S} A_i} w(u)$. Alternatively, coverage functions can be described as non-negative linear combinations of monotone disjunctions. They are a natural subclass of submodular functions and arise in a number of applications. We give an algorithm that for any $γ,δ>0$, given random and uniform…
Minimax Optimality and Spectral Routing for Majority-Vote Ensembles under Markov Dependence
Department of Computer Science, Iowa State University（爱荷华州立大学计算机科学系）；Department of Civil, Construction & Environmental Engineering, Iowa State University（爱荷华州立大学土木、建设与环境工程系）
