The rapidly emerging field of computational pathology has the potential to enable objective diagnosis, therapeutic response prediction and identification of new morphological features of clinical relevance. However, deep learning-based computational pathology approaches either require manual annotation of gigapixel whole slide images (WSIs) in fully-supervised settings or thousands of WSIs with slide-level labels in a weakly-supervised setting. Moreover, whole slide level computational pathology methods also suffer from domain adaptation and interpretability issues. These challenges have prevented the broad adaptation of computational pathology for clinical and research purposes. Here we present CLAM - Clustering-constrained attention multiple instance learning, an easy-to-use, high-throughput, and interpretable WSI-level processing and learning method that only requires slide-level labels while being data efficient, adaptable and capable of handling multi-class subtyping problems. CLAM is a deep-learning-based weakly-supervised method that uses attention-based learning to automatically identify sub-regions of high diagnostic value in order to accurately classify the whole slide, while also utilizing instance-level clustering over the representative regions identified to constrain and refine the feature space. In three separate analyses, we demonstrate the data efficiency and adaptability of CLAM and its superior performance over standard weakly-supervised classification. We demonstrate that CLAM models are interpretable and can be used to identify well-known and new morphological features. We further show that models trained using CLAM are adaptable to independent test cohorts, cell phone microscopy images, and biopsies. CLAM is a general-purpose and adaptable method that can be used for a variety of different computational pathology tasks in both clinical and research settings.

Data Efficient and Weakly Supervised Computational Pathology on Whole Slide Images

Deep-learning methods for computational pathology require either manual annotation of gigapixel whole-slide images (WSIs) or large datasets of WSIs with slide-level labels and typically suffer from poor domain adaptation and interpretability. Here we report an interpretable weakly supervised deep-learning method for data-efficient WSI processing and learning that only requires slide-level labels. The method, which we named clustering-constrained-attention multiple-instance learning (CLAM), uses attention-based learning to identify subregions of high diagnostic value to accurately classify whole slides and instance-level clustering over the identified representative regions to constrain and refine the feature space. By applying CLAM to the subtyping of renal cell carcinoma and non-small-cell lung cancer as well as the detection of lymph node metastasis, we show that it can be used to localize well-known morphological features on WSIs without the need for spatial labels, that it overperforms standard weakly supervised classification algorithms and that it is adaptable to independent test cohorts, smartphone microscopy and varying tissue content.

Data-efficient and weakly supervised computational pathology on whole-slide images.

Recent works have shown that stochastic gradient descent (SGD) achieves the fast convergence rates of full-batch gradient descent for over-parameterized models satisfying certain interpolation conditions. However, the step-size used in these works depends on unknown quantities and SGD's practical performance heavily relies on the choice of this step-size. We propose to use line-search techniques to automatically set the step-size when training models that can interpolate the data. In the interpolation setting, we prove that SGD with a stochastic variant of the classic Armijo line-search attains the deterministic convergence rates for both convex and strongly-convex functions. Under additional assumptions, SGD with Armijo line-search is shown to achieve fast convergence for non-convex functions. Furthermore, we show that stochastic extra-gradient with a Lipschitz line-search attains linear convergence for an important class of non-convex functions and saddle-point problems satisfying interpolation. To improve the proposed methods' practical performance, we give heuristics to use larger step-sizes and acceleration. We compare the proposed algorithms against numerous optimization methods on standard classification tasks using both kernel methods and deep networks. The proposed methods result in competitive performance across all models and datasets, while being robust to the precise choices of hyper-parameters. For multi-class classification using deep networks, SGD with Armijo line-search results in both faster convergence and better generalization.

Painless Stochastic Gradient: Interpolation, Line-Search, and Convergence Rates

Choosing the optimizer is considered to be among the most crucial design decisions in deep learning, and it is not an easy one. The growing literature now lists hundreds of optimization methods. In the absence of clear theoretical guidance and conclusive empirical evidence, the decision is often made based on anecdotes. In this work, we aim to replace these anecdotes, if not with a conclusive ranking, then at least with evidence-backed heuristics. To do so, we perform an extensive, standardized benchmark of fifteen particularly popular deep learning optimizers while giving a concise overview of the wide range of possible choices. Analyzing more than $50,000$ individual runs, we contribute the following three points: (i) Optimizer performance varies greatly across tasks. (ii) We observe that evaluating multiple optimizers with default parameters works approximately as well as tuning the hyperparameters of a single, fixed optimizer. (iii) While we cannot discern an optimization method clearly dominating across all tested tasks, we identify a significantly reduced subset of specific optimizers and parameter choices that generally lead to competitive results in our experiments: Adam remains a strong contender, with newer methods failing to significantly and consistently outperform it. Our open-sourced results are available as challenging and well-tuned baselines for more meaningful evaluations of novel optimization methods without requiring any further computational efforts.

Descending through a Crowded Valley - Benchmarking Deep Learning Optimizers

We propose a stochastic variant of the classical Polyak step-size (Polyak, 1987) commonly used in the subgradient method. Although computing the Polyak step-size requires knowledge of the optimal function values, this information is readily available for typical modern machine learning applications. Consequently, the proposed stochastic Polyak step-size (SPS) is an attractive choice for setting the learning rate for stochastic gradient descent (SGD). We provide theoretical convergence guarantees for SGD equipped with SPS in different settings, including strongly convex, convex and non-convex functions. Furthermore, our analysis results in novel convergence guarantees for SGD with a constant step-size. We show that SPS is particularly effective when training over-parameterized models capable of interpolating the training data. In this setting, we prove that SPS enables SGD to converge to the true solution at a fast rate without requiring the knowledge of any problem-dependent constants or additional computational overhead. We experimentally validate our theoretical results via extensive experiments on synthetic and real datasets. We demonstrate the strong performance of SGD with SPS compared to state-of-the-art optimization methods when training over-parameterized models.

Stochastic Polyak Step-size for SGD: An Adaptive Learning Rate for Fast Convergence

Differential Privacy (DP) provides a formal privacy guarantee preventing adversaries with access to a machine learning model from extracting information about individual training points. Differentially Private Stochastic Gradient Descent (DP-SGD), the most popular DP training method for deep learning, realizes this protection by injecting noise during training. However previous works have found that DP-SGD often leads to a significant degradation in performance on standard image classification benchmarks. Furthermore, some authors have postulated that DP-SGD inherently performs poorly on large models, since the norm of the noise required to preserve privacy is proportional to the model dimension. In contrast, we demonstrate that DP-SGD on over-parameterized models can perform significantly better than previously thought. Combining careful hyper-parameter tuning with simple techniques to ensure signal propagation and improve the convergence rate, we obtain a new SOTA without extra data on CIFAR-10 of 81.4% under (8, 10^{-5})-DP using a 40-layer Wide-ResNet, improving over the previous SOTA of 71.7%. When fine-tuning a pre-trained NFNet-F3, we achieve a remarkable 83.8% top-1 accuracy on ImageNet under (0.5, 8*10^{-7})-DP. Additionally, we also achieve 86.7% top-1 accuracy under (8, 8 \cdot 10^{-7})-DP, which is just 4.3% below the current non-private SOTA for this task. We believe our results are a significant step towards closing the accuracy gap between private and non-private image classification.

Unlocking High-Accuracy Differentially Private Image Classification through Scale

The top-k error is a common measure of performance in machine learning and computer vision. In practice, top-k classification is typically performed with deep neural networks trained with the cross-entropy loss. Theoretical results indeed suggest that cross-entropy is an optimal learning objective for such a task in the limit of infinite data. In the context of limited and noisy data however, the use of a loss function that is specifically designed for top-k classification can bring significant improvements. Our empirical evidence suggests that the loss function must be smooth and have non-sparse gradients in order to work well with deep neural networks. Consequently, we introduce a family of smoothed loss functions that are suited to top-k optimization via deep learning. The widely used cross-entropy is a special case of our family. Evaluating our smooth loss functions is computationally challenging: a na\"ive algorithm would require $\mathcal{O}(\binom{n}{k})$ operations, where n is the number of classes. Thanks to a connection to polynomial algebra and a divide-and-conquer approach, we provide an algorithm with a time complexity of $\mathcal{O}(k n)$. Furthermore, we present a novel approximation to obtain fast and stable algorithms on GPUs with single floating point precision. We compare the performance of the cross-entropy loss and our margin-based losses in various regimes of noise and data size, for the predominant use case of k=5. Our investigation reveals that our loss is more robust to noise and overfitting than cross-entropy.

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Smooth Loss Functions for Deep Top-k Classification

The top-k error is a common measure of performance in machine learning and computer vision In practice, top-k classification is typically performed with deep neural networks trained with the cross-entropy loss Theoretical results indeed suggest that cross-entropy is an optimal learning objective for such a task in the limit of infinite data In the context of limited and noisy data however, the use of a loss function that is specifically designed for top-k classification can bring significant improvements Our empirical evidence suggests that the loss function must be smooth and have non-sparse gradients in order to work well with deep neural networks Consequently, we introduce a family of smoothed loss functions that are suited to top-k optimization via deep learning The widely used cross-entropy is a special case of our family Evaluating our smooth loss functions is computationally challenging: a na\"ive algorithm would require $\mathcal{O}(\binom{n}{k})$ operations, where n is the number of classes Thanks to a connection to polynomial algebra and a divide-and-conquer approach, we provide an algorithm with a time complexity of $\mathcal{O}(k n)$ Furthermore, we present a novel approximation to obtain fast and stable algorithms on GPUs with single floating point precision We compare the performance of the cross-entropy loss and our margin-based losses in various regimes of noise and data size, for the predominant use case of k=5 Our investigation reveals that our loss is more robust to noise and overfitting than cross-entropy

Smooth Loss Functions for Deep Top-k Classification.

The majority of modern deep learning models are able to interpolate the data: the empirical loss can be driven near zero on all samples simultaneously. In this work, we explicitly exploit this interpolation property for the design of a new optimization algorithm for deep learning. Specifically, we use it to compute an adaptive learning-rate given a stochastic gradient direction. This results in the Adaptive Learning-rates for Interpolation with Gradients (ALI-G) algorithm. ALI-G retains the advantages of SGD, which are low computational cost and provable convergence in the convex setting. But unlike SGD, the learning-rate of ALI-G can be computed inexpensively in closed-form and does not require a manual schedule. We provide a detailed analysis of ALI-G in the stochastic convex setting with explicit convergence rates. In order to obtain good empirical performance in deep learning, we extend the algorithm to use a maximal learning-rate, which gives a single hyper-parameter to tune. We show that employing such a maximal learning-rate has an intuitive proximal interpretation and preserves all convergence guarantees. We provide experiments on a variety of architectures and tasks: (i) learning a differentiable neural computer; (ii) training a wide residual network on the SVHN data set; (iii) training a Bi-LSTM on the SNLI data set; and (iv) training wide residual networks and densely connected networks on the CIFAR data sets. We empirically show that ALI-G outperforms adaptive gradient methods such as Adam, and provides comparable performance with SGD, although SGD benefits from manual learning rate schedules. We release PyTorch and Tensorflow implementations of ALI-G as standalone optimizers that can be used as a drop-in replacement in existing code (code available at this https URL ).

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Training Neural Networks for and by Interpolation

Learning a deep neural network requires solving a challenging optimization problem: it is a high-dimensional, non-convex and non-smooth minimization problem with a large number of terms. The current practice in neural network optimization is to rely on the stochastic gradient descent (SGD) algorithm or its adaptive variants. However, SGD requires a hand-designed schedule for the learning rate. In addition, its adaptive variants tend to produce solutions that generalize less well on unseen data than SGD with a hand-designed schedule. We present an optimization method that offers empirically the best of both worlds: our algorithm yields good generalization performance while requiring only one hyper-parameter. Our approach is based on a composite proximal framework, which exploits the compositional nature of deep neural networks and can leverage powerful convex optimization algorithms by design. Specifically, we employ the Frank-Wolfe (FW) algorithm for SVM, which computes an optimal step-size in closed-form at each time-step. We further show that the descent direction is given by a simple backward pass in the network, yielding the same computational cost per iteration as SGD. We present experiments on the CIFAR and SNLI data sets, where we demonstrate the significant superiority of our method over Adam, Adagrad, as well as the recently proposed BPGrad and AMSGrad. Furthermore, we compare our algorithm to SGD with a hand-designed learning rate schedule, and show that it provides similar generalization while converging faster. The code is publicly available at this https URL.

Leonard Berrada

Papers

Unlocking High-Accuracy Differentially Private Image Classification through Scale

Smooth Loss Functions for Deep Top-k Classification

Smooth Loss Functions for Deep Top-k Classification.

Training Neural Networks for and by Interpolation

Deep Frank-Wolfe For Neural Network Optimization