Recent progress was made in characterizing the generalization error of gradient methods for general convex loss by the learning theory community. In this work, we focus on how training longer might affect generalization in smooth stochastic convex optimization (SCO) problems. We first provide tight lower bounds for general non-realizable SCO problems. Furthermore, existing upper bound results suggest that sample complexity can be improved by assuming the loss is realizable, i.e. an optimal solution simultaneously minimizes all the data points. However, this improvement is compromised when training time is long and lower bounds are lacking. Our paper examines this observation by providing excess risk lower bounds for gradient descent (GD) and stochastic gradient descent (SGD) in two realizable settings: 1) realizable with $T = O(n)$, and (2) realizable with $T = \Omega(n)$, where $T$ denotes the number of training iterations and $n$ is the size of the training dataset. These bounds are novel and informative in characterizing the relationship between $T$ and $n$. In the first small training horizon case, our lower bounds almost tightly match and provide the first optimal certificates for the corresponding upper bounds. However, for the realizable case with $T = \Omega(n)$, a gap exists between the lower and upper bounds. We provide a conjecture to address this problem, that the gap can be closed by improving upper bounds, which is supported by our analyses in one-dimensional and linear regression scenarios.
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We consider alternating gradient descent (AGD) with fixed step size $\eta > 0$, applied to the asymmetric matrix factorization objective. We show that, for a rank-$r$ matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$, $T = \left( \left(\frac{\sigma_1(\mathbf{A})}{\sigma_r(\mathbf{A})}\right)^2 \log(1/\epsilon)\right)$ iterations of alternating gradient descent suffice to reach an $\epsilon$-optimal factorization $\| \mathbf{A} - \mathbf{X}_T^{\vphantom{\intercal}} \mathbf{Y}_T^{\intercal} \|_{\rm F}^2 \leq \epsilon \| \mathbf{A} \|_{\rm F}^2$ with high probability starting from an atypical random initialization. The factors have rank $d>r$ so that $\mathbf{X}_T\in\mathbb{R}^{m \times d}$ and $\mathbf{Y}_T \in\mathbb{R}^{n \times d}$. Experiments suggest that our proposed initialization is not merely of theoretical benefit, but rather significantly improves convergence of gradient descent in practice. Our proof is conceptually simple: a uniform PL-inequality and uniform Lipschitz smoothness constant are guaranteed for a sufficient number of iterations, starting from our random initialization. Our proof method should be useful for extending and simplifying convergence analyses for a broader class of nonconvex low-rank factorization problems.
This work studies the generalization error of gradient methods. More specifically, we focus on how training steps $T$ and step-size $\eta$ might affect generalization in smooth stochastic convex optimization (SCO) problems. We first provide tight excess risk lower bounds for Gradient Descent (GD) and Stochastic Gradient Descent (SGD) under the general non-realizable smooth SCO setting, suggesting that existing stability analyses are tight in step-size and iteration dependence, and that overfitting provably happens. Next, we study the case when the loss is realizable, i.e. an optimal solution minimizes all the data points. Recent works show better rates can be attained but the improvement is reduced when training time is long. Our paper examines this observation by providing excess risk lower bounds for GD and SGD in two realizable settings: 1) $\eta T = \bigO{n}$, and (2) $\eta T = \bigOmega{n}$, where $n$ is the size of dataset. In the first case $\eta T = \bigOmega{n}$, our lower bounds tightly match and certify the respective upper bounds. However, for the case $\eta T = \bigOmega{n}$, our analysis indicates a gap between the lower and upper bounds. A conjecture is proposed that the gap can be closed by improving upper bounds, supported by analyses in two special scenarios.
Likelihood-free inference methods typically make use of a distance between simulated and real data. A common example is the maximum mean discrepancy (MMD), which has previously been used for approximate Bayesian computation, minimum distance estimation, generalised Bayesian inference, and within the nonparametric learning framework. The MMD is commonly estimated at a root-$m$ rate, where $m$ is the number of simulated samples. This can lead to significant computational challenges since a large $m$ is required to obtain an accurate estimate, which is crucial for parameter estimation. In this paper, we propose a novel estimator for the MMD with significantly improved sample complexity. The estimator is particularly well suited for computationally expensive smooth simulators with low- to mid-dimensional inputs. This claim is supported through both theoretical results and an extensive simulation study on benchmark simulators.
In this paper, we present a novel stochastic normal map-based algorithm ($\mathsf{norM}\text{-}\mathsf{SGD}$) for nonconvex composite-type optimization problems and discuss its convergence properties. Using a time window-based strategy, we first analyze the global convergence behavior of $\mathsf{norM}\text{-}\mathsf{SGD}$ and it is shown that every accumulation point of the generated sequence of iterates $\{\boldsymbol{x}^k\}_k$ corresponds to a stationary point almost surely and in an expectation sense. The obtained results hold under standard assumptions and extend the more limited convergence guarantees of the basic proximal stochastic gradient method. In addition, based on the well-known Kurdyka-{\L}ojasiewicz (KL) analysis framework, we provide novel point-wise convergence results for the iterates $\{\boldsymbol{x}^k\}_k$ and derive convergence rates that depend on the underlying KL exponent $\boldsymbol{\theta}$ and the step size dynamics $\{\alpha_k\}_k$. Specifically, for the popular step size scheme $\alpha_k=\mathcal{O}(1/k^\gamma)$, $\gamma \in (\frac23,1]$, (almost sure) rates of the form $\|\boldsymbol{x}^k-\boldsymbol{x}^*\| = \mathcal{O}(1/k^p)$, $p \in (0,\frac12)$, can be established. The obtained rates are faster than related and existing convergence rates for $\mathsf{SGD}$ and improve on the non-asymptotic complexity bounds for $\mathsf{norM}\text{-}\mathsf{SGD}$.
Conventionally, federated learning aims to optimize a single objective, typically the utility. However, for a federated learning system to be trustworthy, it needs to simultaneously satisfy multiple/many objectives, such as maximizing model performance, minimizing privacy leakage and training cost, and being robust to malicious attacks. Multi-Objective Optimization (MOO) aiming to optimize multiple conflicting objectives at the same time is quite suitable for solving the optimization problem of Trustworthy Federated Learning (TFL). In this paper, we unify MOO and TFL by formulating the problem of constrained multi-objective federated learning (CMOFL). Under this formulation, existing MOO algorithms can be adapted to TFL straightforwardly. Different from existing CMOFL works focusing on utility, efficiency, fairness, and robustness, we consider optimizing privacy leakage along with utility loss and training cost, the three primary objectives of a TFL system. We develop two improved CMOFL algorithms based on NSGA-II and PSL, respectively, for effectively and efficiently finding Pareto optimal solutions, and we provide theoretical analysis on their convergence. We design specific measurements of privacy leakage, utility loss, and training cost for three privacy protection mechanisms: Randomization, BatchCrypt (An efficient version of homomorphic encryption), and Sparsification. Empirical experiments conducted under each of the three protection mechanisms demonstrate the effectiveness of our proposed algorithms.
We propose policy gradient algorithms for solving a risk-sensitive reinforcement learning problem in on-policy as well as off-policy settings. We consider episodic Markov decision processes, and model the risk using the broad class of smooth risk measures of the cumulative discounted reward. We propose two template policy gradient algorithms that optimize a smooth risk measure in on-policy and off-policy RL settings, respectively. We derive non-asymptotic bounds that quantify the rate of convergence to our proposed algorithms to a stationary point of the smooth risk measure. As special cases, we establish that our algorithms apply to the optimization of mean-variance and distortion risk measures, respectively.
Macro-AUC is the arithmetic mean of the class-wise AUCs in multi-label learning and is commonly used in practice. However, its theoretical understanding is far lacking. Toward solving it, we characterize the generalization properties of various learning algorithms based on the corresponding surrogate losses w.r.t. Macro-AUC. We theoretically identify a critical factor of the dataset affecting the generalization bounds: \emph{the label-wise class imbalance}. Our results on the imbalance-aware error bounds show that the widely-used univariate loss-based algorithm is more sensitive to the label-wise class imbalance than the proposed pairwise and reweighted loss-based ones, which probably implies its worse performance. Moreover, empirical results on various datasets corroborate our theory findings. To establish it, technically, we propose a new (and more general) McDiarmid-type concentration inequality, which may be of independent interest.
We consider problems of minimizing functionals $\mathcal{F}$ of probability measures on the Euclidean space. To propose an accelerated gradient descent algorithm for such problems, we consider gradient flow of transport maps that give push-forward measures of an initial measure. Then we propose a deterministic accelerated algorithm by extending Nesterov's acceleration technique with momentum. This algorithm do not based on the Wasserstein geometry. Furthermore, to estimate the convergence rate of the accelerated algorithm, we introduce new convexity and smoothness for $\mathcal{F}$ based on transport maps. As a result, we can show that the accelerated algorithm converges faster than a normal gradient descent algorithm. Numerical experiments support this theoretical result.
It is often desirable to summarise a probability measure on a space $X$ in terms of a mode, or MAP estimator, i.e.\ a point of maximum probability. Such points can be rigorously defined using masses of metric balls in the small-radius limit. However, the theory is not entirely straightforward: the literature contains multiple notions of mode and various examples of pathological measures that have no mode in any sense. Since the masses of balls induce natural orderings on the points of $X$, this article aims to shed light on some of the problems in non-parametric MAP estimation by taking an order-theoretic perspective, which appears to be a new one in the inverse problems community. This point of view opens up attractive proof strategies based upon the Cantor and Kuratowski intersection theorems; it also reveals that many of the pathologies arise from the distinction between greatest and maximal elements of an order, and from the existence of incomparable elements of $X$, which we show can be dense in $X$, even for an absolutely continuous measure on $X = \mathbb{R}$.
Most state-of-the-art machine learning techniques revolve around the optimisation of loss functions. Defining appropriate loss functions is therefore critical to successfully solving problems in this field. We present a survey of the most commonly used loss functions for a wide range of different applications, divided into classification, regression, ranking, sample generation and energy based modelling. Overall, we introduce 33 different loss functions and we organise them into an intuitive taxonomy. Each loss function is given a theoretical backing and we describe where it is best used. This survey aims to provide a reference of the most essential loss functions for both beginner and advanced machine learning practitioners.
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