From 1d3d27cfe00fc5a849c7768e469b1a2ade537943 Mon Sep 17 00:00:00 2001 From: David Emerson <43939939+emersodb@users.noreply.github.com> Date: Wed, 14 May 2025 18:49:34 -0400 Subject: [PATCH 1/8] Two initial references to check in. Two more to go --- books/fl/src/SUMMARY.md | 8 +- books/fl/src/assets/algorithm-fedopt.svg | 1010 +++++++++++++++++ books/fl/src/assets/fed_df_model.png | Bin 0 -> 108166 bytes books/fl/src/assets/fedavg_model.png | Bin 0 -> 146524 bytes .../assets/heterogeneity_two_routes_alt.svg | 4 + books/fl/src/assets/local_model_1.png | Bin 0 -> 120874 bytes books/fl/src/assets/local_model_2.png | Bin 0 -> 91246 bytes books/fl/src/horizontal/README.md | 2 +- .../src/horizontal/robust_global_fl/README.md | 173 +++ .../horizontal/robust_global_fl/fedadam.md | 1 - .../src/horizontal/robust_global_fl/fedopt.md | 131 +++ books/fl/src/horizontal/vanilla_fl/fedavg.md | 2 +- books/fl/src/horizontal/vanilla_fl/fedsgd.md | 2 +- 13 files changed, 1325 insertions(+), 8 deletions(-) create mode 100644 books/fl/src/assets/algorithm-fedopt.svg create mode 100644 books/fl/src/assets/fed_df_model.png create mode 100644 books/fl/src/assets/fedavg_model.png create mode 100644 books/fl/src/assets/heterogeneity_two_routes_alt.svg create mode 100644 books/fl/src/assets/local_model_1.png create mode 100644 books/fl/src/assets/local_model_2.png delete mode 100644 books/fl/src/horizontal/robust_global_fl/fedadam.md create mode 100644 books/fl/src/horizontal/robust_global_fl/fedopt.md diff --git a/books/fl/src/SUMMARY.md b/books/fl/src/SUMMARY.md index 0fdd43d..f1baa3f 100644 --- a/books/fl/src/SUMMARY.md +++ b/books/fl/src/SUMMARY.md @@ -20,10 +20,10 @@ - [Vanilla FL](horizontal/vanilla_fl/README.md) - [FedSGD](horizontal/vanilla_fl/fedsgd.md) - [FedAvg](horizontal/vanilla_fl/fedavg.md) - - [Robust Global FL]() <-- (horizontal/robust_global_fl/README.md) --> - - [FedAdam]() <-- (horizontal/robust_global_fl/fedadam.md) --> - - [FedProx]() <-- (horizontal/robust_global_fl/fedprox.md) --> - - [MOON]() <-- (horizontal/robust_global_fl/moon.md) --> + - [Robust Global FL](horizontal/robust_global_fl/README.md) + - [FedOpt](horizontal/robust_global_fl/fedopt.md) + - [FedProx](horizontal/robust_global_fl/fedprox.md) + - [MOON](horizontal/robust_global_fl/moon.md) - [Personalized FL]() <-- (horizontal/personalized/README.md) --> - [FedPer]() <-- (horizontal/personalized/fedper.md) --> - [FENDA-FL]() <-- (horizontal/personalized/fenda.md) --> diff --git a/books/fl/src/assets/algorithm-fedopt.svg b/books/fl/src/assets/algorithm-fedopt.svg new file mode 100644 index 0000000..ce22cd6 --- /dev/null +++ b/books/fl/src/assets/algorithm-fedopt.svg @@ -0,0 +1,1010 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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Global FL with Robust Optimization
Personalized FL
Two Common
Routes
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dzdFEEl@1%f7rp!Zpi7iPSJObFLhX6P{{gf^an=9; literal 0 HcmV?d00001 diff --git a/books/fl/src/horizontal/README.md b/books/fl/src/horizontal/README.md index ada2d5f..e18172c 100644 --- a/books/fl/src/horizontal/README.md +++ b/books/fl/src/horizontal/README.md @@ -54,7 +54,7 @@ This section of the book is organized as follows: - [FedSGD](vanilla_fl/fedsgd.md) - [FedAvg](vanilla_fl/fedavg.md) - [Robust Global FL](robust_global_fl/index.md) - - [FedAdam](robust_global_fl/fedadam.md) + - [FedOpt](robust_global_fl/fedopt.md) - [FedProx](robust_global_fl/fedprox.md) - [MOON](robust_global_fl/moon.md) - [Personalized FL](personalized/index.md) diff --git a/books/fl/src/horizontal/robust_global_fl/README.md b/books/fl/src/horizontal/robust_global_fl/README.md index 64427d4..a0b8f0a 100644 --- a/books/fl/src/horizontal/robust_global_fl/README.md +++ b/books/fl/src/horizontal/robust_global_fl/README.md @@ -1 +1,174 @@ + + # Robust Global FL Approaches + +{{ #aipr_header }} + +## Data heterogeneity in standard ML + +In standard ML, when training and deploying a model, a standard underlying +assumption assumption is that the training data is distributionally +similar to new data to which the model will be applied. There are methods +that specialize in out-of-domain generalization, but, in most cases, trained +models are assumed to be applied on data that is drawn from the same +statistical distributions that describe the data on which it was trained. The +validity of this assumption can degrade, for example, over time or due to +the model being used to make predictions in entirely new domains. + +While data shifts present a significant challenge in centralized ML training, +the characteristics that describe data shifts in this domain also exist in FL +when comparing disparate, distributed datasets. Data shift between such +datasets is typically referred to as "data heterogeneity" between clients. Such +heterogeneity introduces new obstacles in FL and is quite prevalent. Before +discussing its impact on federated training and how it is addressed. Let's +define some types of data divergence. Three common ways to describe +disparities or shifts between the training and inference data are:[^1] + +1. Label Shift +2. Covariate Shift +3. Concept Drift + +Let \\(X\\) and \\(Y\\) represent the feature (input) and label (output) +spaces, respectively for a model. Shifts are present, regardless if model +performance degrades, when the joint distributions + +$$ +\begin{align} +P\_{\\text{train}}(X, Y) \\neq P\_{\\text{test}}(X, Y). \tag{1} +\end{align} +$$ + +### Label Shift + +Label shifts occur when there is a change in the label distribution \\(\\mathbb{P}(Y)\\) +with a fixed posterior distribution \\(\\mathbb{P}(X \\vert Y)\\). That is, the +probability of seeing different label values shifts, but the distribution of +features conditioned on the labels does not change. A pertinent example of +this might be data meant to train a model to diagnose COVID-19 in the early +days of spread versus the later stages when the virus was widely circulating. +Generally, the symptoms, given that someone had the virus, did not markedly +change. However, the prevalence of the virus, \\(\\mathbb{P}(Y)\\), did. + +### Covariate Shift + +Covariate shifts between data distributions represent a change in the feature +distribution, \\(\\mathbb{P}(X)\\), while the statistical relationship of labels to +features, \\(\\mathbb{P}(Y \\vert X)\\), remains fixed. Consider the setting of training +a readmission risk model on data drawn from the patient population of a +general hospital. If, for instance, that model were transferred for use at a +nearby pediatric hospital. Assuming all else equal, predictions from that model +would be influenced by covariate drift due to the change in patient +demographics. Namely, though features associated with younger patients are +likely part of the general hospital population, they will, of course, be +statistically over-represented in the data points seen by the model at the +pediatric hospital. + +### Concept Drift + +Concept drift is characterized by a change in \\(\\mathbb{P}(Y \vert X)\\) provided a +fixed \\(\\mathbb{P}(Y)\\). Essentially, this drift encapsulates a shift in the +predictive relationship between the features, \\(X\\), and the labels, \\(Y\\). +As an illustrative example, consider training a purchase conversion model where +for airline tickets where two possible incentives are features. The first +offers a ticket discount to encourage purchase, whereas the second offers free +add-ons. In good economic periods, the second incentive may produce higher +conversion rates. On the other hand, in periods of economic uncertainty, +perhaps the first offer would do so. + +Note that each of the shifts discussed above may exist in isolation or be +present together in varying degrees. + +## How does data heterogeneity manifest in FL? + +In FL, differences in training data distributions are not strictly temporal or +marked by a change in the joint probability distributions of the training and +test datasets, as expressed in Equation (1). Each client participating in +federated training might naturally exhibit distribution disparities. Consider +the example given in the Section on [Covariate Shift](#covariate-shift). If +the general and pediatric hospitals would like to collaboratively train a model +using FL, the demographics of their patient population mean that there will be +substantial statistical heterogeneity in their respective training datasets. + +Each distributed training dataset in an FL system may naturally exhibit the +various disparities, compared with one another, discussed above. As a further +example, consider two financial institutions working together to train an fraud +detection model. Because of their different clientele, one bank may experience +fraud at a rate of 2% per transaction, while the other may see only 0.1%. + +## How does it impact FL models? + +Data heterogeneity, in its various forms, has been linked to a number of +challenges in training FL models using methods like +[FedAvg](../vanilla_fl/fedavg.md), including slower convergence, performance +degradation, and unevenly distributed training dynamics among clients. In [2], +a clear illustration of the impact of data heterogeneity is provided. In the +figures below, two clients have locally trained a model on their respective +datasets. + +

+
+Local Model 1 +Local Model 2 +
Two clients with different datasets. Note that each holds a +slightly different view of the feature space. Notably, Client 1 (left) has a +distinct cluster of data points in the bottom right and fewer points labeled in +green within the red cluster.
+
+
+ +The decision boundaries of the locally trained models are largely similar but +differ in important ways. If the two models are averaged via FedAvg (see figure +below), the result is a blurred decision boundary which has diverged from the +sharp boundary one would expect to compute were the data agglomerated and a +central model trained. Alternatively, using an approach that is more robust +to data heterogeneity, FedDF,[^2] the resulting model exhibits the kinds of +classification boundaries one would expect when considering the data +distributions from a global perspective. + +
+
+FedAvg Model +FedDF Model +
Model resulting from FedAvg (left) compared with the model +trained using FedDF[^2] (right).
+
+
+ +There are two common routes, among many other routes, for addressing +heterogeneity in FL. The first is to maintain a sense of a single global to be +trained by all participants. Modifications to items like the aggregation +strategy, local learning objectives, or corrections to model updates are +applied to better align FL training with the dynamics of centralized training +without sacrificing most of the benefits associated with the original FedAvg +algorithm. The second route is to abandon, to one degree or another, the idea +of a global model that performs well across all clients and instead allow +each client to train a unique model. This is known as Personal or Personalized +FL (pFL). Such models still benefit from global information through aspects of +FL, but more strongly emphasize local distributions. + +
+
+Two FL Routes +
Two possible routes for addressing data heterogeneity in FL.
+
+
+ +In the subsequent sections of this chapter, we'll cover a few of the many FL +methods aimed at robust global model optimization in FL. Such models are often +more generalizable and are more easily distributed to new domains than their +pFL equivalents. Alternatively, model performance on each client may not be +as high as those produced by pFL approaches. + +#### References & Useful Links + +[^1]: + J. Quinonero-Candela, M. Sugiyama, A. Schwaighofer, and N. D. Lawrence. Dataset shift + in machine learning. Mit Press, 2008 + +[^2]: + [Lin, Tao et al. “Ensemble distillation for robust model fusion in federated + learning”. In: Proceedings of the 34th International Conference on Neural + Information Processing Systems. NIPS ’20. Vancouver, BC, Canada: Curran + Associates Inc., 2020.](https://proceedings.neurips.cc/paper/2020/file/18df51b97ccd68128e994804f3eccc87-Paper.pdf) + +{{#author emersodb}} diff --git a/books/fl/src/horizontal/robust_global_fl/fedadam.md b/books/fl/src/horizontal/robust_global_fl/fedadam.md deleted file mode 100644 index 166cea9..0000000 --- a/books/fl/src/horizontal/robust_global_fl/fedadam.md +++ /dev/null @@ -1 +0,0 @@ -# FedAdam diff --git a/books/fl/src/horizontal/robust_global_fl/fedopt.md b/books/fl/src/horizontal/robust_global_fl/fedopt.md new file mode 100644 index 0000000..d51a3b6 --- /dev/null +++ b/books/fl/src/horizontal/robust_global_fl/fedopt.md @@ -0,0 +1,131 @@ + + +# The FedOpt Family of Aggregation Strategies + +{{ #aipr_header }} + +Recall that modern deep learning optimizers like AdamW[^1] or AdaGrad[^2] use +first- and second-order moment estimates of the stochastic gradients computed +during iterative optimization to adaptively modify the model updates. +At a high level, each algorithm aims to reinforce common update directions +(i.e. those with momentum) and damp update noisy directions (i.e. those with +high batch-to-batch variance). The FedOpt family[^3] of algorithms, considers +modifying the traditional FedAvg aggregation algorithm to incorporate similar +adaptations into server-side model updates in FL. + +## Mathematical motivation + +In [FedAvg](../vanilla_fl/fedavg.md), recall that, after a round of local +training on each client, client model weights are combined into a single model +representation via + +$$ +\begin{align*} +\\mathbf{w}\_{t+1} = \\sum\_{k \\in C_t} \\frac{n_k}{n_s} \\mathbf{w}^k_{t+1}, +\end{align*} +$$ + +where \\(\\mathbf{w}^k\_{t+1}\\) is simply the model weights after local +training on client \\(k\\). For round \\(t\\), each client starts local +training from the same set of weights, \\(\\mathbf{w_t}\\). Assume that each +client has the same number of data points such that \\(n_k = m\\). With a bit +of algebra, the update is rewritten + +$$ +\begin{align} +\\mathbf{w}\_{t+1} = \\sum\_{k \\in C_t} \\frac{n_k}{n_s} \\mathbf{w}^k\_{t+1} &= \\mathbf{w}_t - \\frac{1}{C_t} \\sum\_{k \\in C_t} +\\left( \\mathbf{w}_t - \\mathbf{w}^k\_{t+1} \\right), \\\\ +&= \\mathbf{w}_t - \\frac{1}{C\_t} \\sum\_{k \\in C_t} \\Delta^k\_{t+1}, \\\\ +&= \\mathbf{w}_t - \\Delta\_{t+1}. \tag{1} +\end{align} +$$ + +Here, \\(\\Delta^k\_{t+1} = \\mathbf{w}\_t - \\mathbf{w}^k\_{t+1}\\) is just +the vector pointing from the initial models weights and to those after local +training and \\(\\Delta\_{t+1}\\) is simply the uniform average of these +update vectors. + +Recall that, if each client uses a fixed learning rate, \\(\eta\\), and +performs a single, full gradient update, FedAvg is equivalent to centralized +large-batch SGD. Similarly, in this case, if each client performs on step of +batch SGD with a learning rate of 1.0, then the update in Equation (1) is +equivalent to a batch-SGD update with a learning rate of 1.0 for the +**server**. The "server-side" batch is the union of the batches used on each +client. + +The observation that \\(\Delta\_{t+1}\\) is simply a stochastic gradient +motivates treating these update directions, +\\(\\Delta^k\_{t+1} = \\mathbf{w}\_t - \\mathbf{w}^k\_{t+1}\\) like the +stochastic gradients in standard adaptive optimizers. It's important to note +that if the clients, for instance, apply multiple steps of local SGD or use +different learning rates, the exact equivalence of \\(\Delta\_{t+1}\\) to a +stochastic gradient is broken. However, it shares similarities to such a +gradient and is, therefore, called a "pseudo-gradient."[^3] + +## The algorithms: FedAdagrad, FedAdam, FedYogi + +Drawing inspiration from three successful, traditional adaptive optimizers, +adaptive server-side aggregation strategy FedAdaGrad, FedAdam, and FedYogi have +been proposed. See the algorithm below for details. + +
+
+FedOpt Algorithms +
+
+ +Those familiar with the mathematical formulations of Adagrad, Adam[^4], and +Yogi[^5] will recognize the general structure of these equations. Computation +of \\(m_t\\), based on the average of the update directions suggested by each +client through local training (\\(\Delta\_{t+1}\\)) serves to accumulate +momentum associated with directions that are consistently and frequently part +of these updates. On the other hand, \\(\nu_t\\) estimates the variance +associated with update directions throughout the server rounds. Directions +with higher variance values are damped in favor of those with more consistency +round over round. + +As with the usual forms of these algorithms, there are a number of +hyper-parameters that can be tuned, including \\(\tau, \beta_1,\\) and +\\(\beta_2\\). However, sensible defaults are suggested in the paper such that +\\(\beta_1=0.9\\) and \\(\beta_2=0.99\\). The authors also show that +performance is generally robust to \\(\tau\\). + +A number of experiments show that the proposed FedOpt family of algorithms +can outperform FedAvg, especially in heterogeneous settings. Moreover, these +algorithms, in the experiments of the paper, outperform SCAFFOLD,[^6] a +variance reduction method aimed at improving convergence in the presence of +heterogeneity. A final advantage of the FedOpt family of algorithms is that +they are accompanied by several convergence results showing that, as long as +the variance of the local gradients is not too large, the algorithms converge +properly. + +#### References & Useful Links + +[^1]: + [I. Loshchilov and F. Hutter. Fixing weight decay regularization in ADAM, + 2018.386](https://arxiv.org/pdf/1711.05101) + +[^2]: + [Duchi, J., Hazan, E., & Singer, Y. (2011). Adaptive subgradient methods + for online learning and stochastic optimization. Journal of machine learning research, 12(7).](https://www.jmlr.org/papers/volume12/duchi11a/duchi11a.pdf) + +[^3]: + [S. J. Reddi, Z. Charles, M. Zaheer, Z. Garrett, K. Rush, J. Konêcný, S. Kumar, and H. B. + McMahan. Adaptive federated optimization. In ICLR 2021, 2021.](https://arxiv.org/abs/2003.002950) + +[^4]: + [Kingma, D. P. & Ba, J. (2015). Adam: A Method for Stochastic Optimization. + In Y. Bengio & Y. LeCun (eds.), ICLR (Poster).](https://arxiv.org/pdf/1412.6980) + +[^5]: + [Manzil Zaheer, Sashank J. Reddi, Devendra Sachan, Satyen Kale, and + Sanjiv Kumar. 2018. Adaptive methods for nonconvex optimization. In Proceedings of the 32nd International Conference on Neural Information Processing Systems (NIPS'18). Curran Associates Inc., Red Hook, NY, USA, 9815–9825.](https://proceedings.neurips.cc/paper_files/paper/2018/file/90365351ccc7437a1309dc64e4db32a3-Paper.pdf) + +[^6]: + [S. P. Karimireddy, S. Kale, M. Mohri, S. Reddi, S. Stich, and A. T. Suresh. + SCAFFOLD: Stochastic controlled averaging for federated learning. In + Hal Daumé III and Aarti Singh, editors, Proceedings of the 37th + International Conference on Machine Learning, volume 119 of Proceedings of + Machine Learning Research, pages 5132–5143. PMLR, 13–18 Jul 2020.](https://www.microsoft.com/en-us/research/publication/frustratingly-easy-neural-domain-adaptation/) + +{{#author emersodb}} diff --git a/books/fl/src/horizontal/vanilla_fl/fedavg.md b/books/fl/src/horizontal/vanilla_fl/fedavg.md index 5b6fa56..440fb0d 100644 --- a/books/fl/src/horizontal/vanilla_fl/fedavg.md +++ b/books/fl/src/horizontal/vanilla_fl/fedavg.md @@ -92,7 +92,7 @@ model as described by the weights \\(\mathbf{w}\_T\\).
-FedAvg Algorithm +FedAvg Algorithm
diff --git a/books/fl/src/horizontal/vanilla_fl/fedsgd.md b/books/fl/src/horizontal/vanilla_fl/fedsgd.md index d740092..40dd0f5 100644 --- a/books/fl/src/horizontal/vanilla_fl/fedsgd.md +++ b/books/fl/src/horizontal/vanilla_fl/fedsgd.md @@ -118,7 +118,7 @@ client receives the final model as described by the weights \\(\mathbf{w}\_T\\).
-FedSGD Algorithm +FedSGD Algorithm
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dzdFEEl@1%f7rp!Zpi7iPSJObFLhX6P{{gf^an=9; From e9cb3488a83ad36cda161ba7659699dc48a4a867 Mon Sep 17 00:00:00 2001 From: David Emerson <43939939+emersodb@users.noreply.github.com> Date: Wed, 14 May 2025 18:56:45 -0400 Subject: [PATCH 3/8] height adjustments --- books/fl/src/horizontal/robust_global_fl/README.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/books/fl/src/horizontal/robust_global_fl/README.md b/books/fl/src/horizontal/robust_global_fl/README.md index a0b8f0a..30b2933 100644 --- a/books/fl/src/horizontal/robust_global_fl/README.md +++ b/books/fl/src/horizontal/robust_global_fl/README.md @@ -107,8 +107,8 @@ datasets. -As in the section on [FedAvg](../vanilla_fl/fedavg.md), there is a substantial +As in the section on [FedAvg](../vanilla_fl/fedavg.md)[^4], there is a substantial reduction in communication overhead if each client applies multiple steps of batch SGD, optimizing the local model based on the local loss. We noted, however, that this breaks the equivalence enjoyed, for example, by FedSGD with @@ -151,4 +151,9 @@ well with and without adaptation. [Images adapted from Talk on "Stochastic Controlled Averaging for Federated Learning"](https://docs.google.com/presentation/d/1SYhRC6NMEMJJL2FTGDu5DJeINJBkbwFVGd2Emy43fk0/edit) +[^4]: + [H. B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y Arcas. + Communication-efficient learning of deep networks from decentralized data. + Proceedings of the 20th AISTATS, 2017.](https://proceedings.mlr.press/v54/mcmahan17a/mcmahan17a.pdf) + {{#author emersodb}} diff --git a/books/fl/src/horizontal/robust_global_fl/moon.md b/books/fl/src/horizontal/robust_global_fl/moon.md index e3ceccf..44311ab 100644 --- a/books/fl/src/horizontal/robust_global_fl/moon.md +++ b/books/fl/src/horizontal/robust_global_fl/moon.md @@ -1 +1,149 @@ -# MOON + + +# MOON: Model-Contrastive Federated Learning + +{{ #aipr_header }} + +The MOON[^1] algorithm is built on the same principles as the +[FedProx](./fedprox.md)[^2] approach. That is, it targets limiting +client-specific drift during local training by constraining how heavily local +model updates stray from global models. The fundamental difference is the +way that drift is measured to construct the penalty function. + +## Contrastive Loss: A Brief Interlude + +Before defining the MOON penalty, we need to review contrastive loss and what +it aims to do in general. Say we have three vectors, \\(\\mathbf{z}\\), +\\(\\mathbf{z}\_s\\), \\(\\mathbf{z}\_d \in \mathbb{R}^n\\) and we want to +optimize a model, parameterized by \\(\\mathbf{w}\\), to map from an input, +\\(\\mathbf{x}\\), to a representation, \\(\\mathbf{z}\\), which is **closer** +to \\(\\mathbf{z}\_s\\) and **further** from \\(\\mathbf{z}\_d\\). We define + +$$ +\begin{align*} +\\ell\_{\\text{con}}(\mathbf{x};\\mathbf{w}) = - \\log \\frac{\exp +\\left(\\text{sim}(\\mathbf{z}, \\mathbf{z}\_s) \\tau^{-1} \\right)}{\\exp +\\left(\\text{sim}(\\mathbf{z}, \\mathbf{z}\_s) \\tau^{-1} \\right) + \\exp +\\left(\\text{sim}(\\mathbf{z}, \\mathbf{z}\_d) \\tau^{-1} \\right)}, +\end{align*} +$$ + +where \\(\\text{sim}(\\cdot, \\cdot)\\) is cosine-similarity and +\\(\tau > 0\\) is a temperature. Increasing the similarity of +\\(\\mathbf{z}\\) to \\(\\mathbf{z}\_s\\) **increases** the numerator. Pushing +\\(\\mathbf{z}\\) away from \\(\\mathbf{z}\_d\\) decreases the denominator. +Each of which makes \\(\\ell\_{\\text{con}}(\mathbf{x};\\mathbf{w})\\) +**smaller**. + +Contrastive loss objectives have been widely used to bring latent +representations of information closer together. For example, contrastive +learning is used extensively in CLIP[^3] as a means of pushing image-caption +text pairs closer together, while pushing unrelated image-text pairs further +apart in the CLIP model's representation space. + +## MOON models and an alternative to weight-drift penalties + +As in FedProx, the major contribution of the MOON algorithm is to modify the +local learning objective for each client. As foreshadowed in the previous +section, the modification involves a contrastive loss function. To define this +loss function, MOON first considers splitting the model to be federally trained +into two stages: a feature map followed by a classification module. +Furthermore, at each server round, \\(t\\), it distinguishes between three +models. These models are illustrated in the figure below. + +
+
+Moon models and their latent representation +
The three models important in the computation of MOON's +contrastive loss functions and their latent representations.
+
+
+ +For server round \\(t\\), the model on the left represents the model after +weight aggregation by the server. In the middle is the final model after +local training on Client \\(i\\). The weights, \\(\\mathbf{w}^{t-1}\_i\\), have +been aggregated across participating clients to form \\(\\mathbf{w}\_t\\). +Finally, the model on the right is the one being locally trained on Client +\\(i\\). The output of the feature maps in these models will be used to form +the local contrastive loss for Client \\(i\\). + +Let \\(\\ell_i((\\mathbf{x}, y); \\mathbf{w})\\) denote the local loss function +of Client \\(i\\) for an input, \\(\mathbf{x}\\), and label, \\(y\\), +parameterized by model weights \\(\\mathbf{w}\\). MOON augments this local loss +with the contrastive loss function + +$$ +\begin{align*} +\\ell\_{i, \\text{con}}((\\mathbf{x}, y);\\mathbf{w}^t\_i) = - \\log \\frac{\exp +\\left(\\text{sim}(\\mathbf{z}, \\mathbf{z}\_{\\text{glob}}) \\tau^{-1} +\\right)}{\\exp \\left(\\text{sim}(\\mathbf{z}, \\mathbf{z}\_{\\text{glob}}) +\\tau^{-1} \\right) + \\exp \\left(\\text{sim}(\\mathbf{z}, +\\mathbf{z}\_{\\text{prev}}) \\tau^{-1} \\right)}. +\end{align*} +$$ + +That is, the local objective for Client \\(i\\) is written + +$$ +\begin{align*} +\\ell_i((\\mathbf{x}, y); \\mathbf{w}^t\_i) + +\mu \\ell\_{i, \\text{con}}((\\mathbf{x}, y);\\mathbf{w}^t\_i), +\end{align*} +$$ + +for \\(\mu > 0\\). Note that, when computed over a batch of data, the average +loss over all data points in the batch is computed. + +The idea here is similar to FedProx. In some sense, we +still want to make sure that, during training, the model does not drift too +fair from the global model, as was the case in FedProx. The difference, here, +is that we're applying that constraint in the feature representation space, +rather than directly in the model weights themselves. In the original work, +MOON showed notable improvements over methods like FedProx in heterogeneous +settings. However, it does not always outperform FedProx or even FedAvg.[^4] +As such, there are likely scenarios where MOON is the right approach for FL in +heterogeneous settings, while others might benefit from an alternative +approach. + +## The algorithm + +The MOON algorithm is fairly similar to that of FedProx. Most server-side +aggregation strategy may be applied. However, the algorithm has some additional +memory overhead, as forward passes of three separate models must be run in +order to extract the latent representations of the data points in each training +batch. In the algorithm below, FedAvg is used as the server-side strategy. + +
+
+Moon Algorithms +
+
+ +#### References & Useful Links + +[^1]: + [Q. Li, B. He, and D. Song. Model-contrastive federated learning. In Proceedings of the + IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2021a.](https://openaccess.thecvf.com/content/CVPR2021/papers/Li_Model-Contrastive_Federated_Learning_CVPR_2021_paper.pdf) + +[^2]: + [T. Li, A. K. Sahu, M. Zaheer, M. Sanjabi, A. Talwalkar, and V. Smith. + Federated optimization in heterogeneous networks. In I. Dhillon, + D. Papailiopoulos, and V. Sze, editors, Proceedings of Machine Learning and + Systems, volume 2, pages 429–450, 2020.](https://arxiv.org/pdf/1812.06127) + +[^3]: + [Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, + Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual + models from natural language supervision. In International conference on machine learning, + pages 8748–8763. PMLR, 2021.](https://proceedings.mlr.press/v139/radford21a/radford21a.pdf) + +[^4]: + [Fatemeh Tavakoli, D. B. Emerson, Sana Ayromlou, John Taylor Jewell, + Amrit Krishnan, Yuchong Zhang, Amol Verma, and Fahad Razak. + A comprehensive view of personalized federated learning on heterogeneous + clinical datasets. In Kaivalya Deshpande, Madalina Fiterau, Shalmali Joshi, + Zachary Lipton, Rajesh Ranganath, and Iñigo Urteaga, editors, Proceedings + of the 9th Machine Learning for Healthcare Conference, volume 252 of + Proceedings of Machine Learning Research. PMLR, 16–17 Aug 2024.](https://proceedings.mlr.press/v252/tavakoli24a.html) + +{{#author emersodb}} From 26f83b353bdb1b850b2b9cfe5cfe7fe8e000658e Mon Sep 17 00:00:00 2001 From: David Emerson <43939939+emersodb@users.noreply.github.com> Date: Fri, 16 May 2025 18:08:37 -0400 Subject: [PATCH 6/8] Grammar and proof pass --- books/fl/src/assets/algorithm-fedopt.svg | 28 +++++------ .../src/horizontal/robust_global_fl/README.md | 50 ++++++++++--------- .../src/horizontal/robust_global_fl/fedopt.md | 43 ++++++++-------- .../horizontal/robust_global_fl/fedprox.md | 38 +++++++------- .../src/horizontal/robust_global_fl/moon.md | 28 ++++++----- 5 files changed, 96 insertions(+), 91 deletions(-) diff --git a/books/fl/src/assets/algorithm-fedopt.svg b/books/fl/src/assets/algorithm-fedopt.svg index ce22cd6..938fc65 100644 --- a/books/fl/src/assets/algorithm-fedopt.svg +++ b/books/fl/src/assets/algorithm-fedopt.svg @@ -22,13 +22,13 @@ inkscape:pageopacity="0.0" inkscape:pagecheckerboard="0" inkscape:deskcolor="#d1d1d1" - inkscape:zoom="0.75438514" - inkscape:cx="349.29108" - inkscape:cy="184.25601" - inkscape:window-width="1512" - inkscape:window-height="837" - inkscape:window-x="0" - inkscape:window-y="38" + inkscape:zoom="0.75662879" + inkscape:cx="229.30663" + inkscape:cy="132.82603" + inkscape:window-width="1920" + inkscape:window-height="948" + inkscape:window-x="1512" + inkscape:window-y="25" inkscape:window-maximized="1" inkscape:current-layer="g1" /> FedDF Model
Model resulting from FedAvg (left) compared with the model -trained using FedDF[^2] (right).
+trained using FedDF (right). There are two common routes, among many other routes, for addressing -heterogeneity in FL. The first is to maintain a sense of a single global to be -trained by all participants. Modifications to items like the aggregation +heterogeneity in FL. The first is to maintain a sense of a single global model +to be trained by all participants. Modifications to items like the aggregation strategy, local learning objectives, or corrections to model updates are applied to better align FL training with the dynamics of centralized training without sacrificing most of the benefits associated with the original FedAvg diff --git a/books/fl/src/horizontal/robust_global_fl/fedopt.md b/books/fl/src/horizontal/robust_global_fl/fedopt.md index d51a3b6..ee1cdf7 100644 --- a/books/fl/src/horizontal/robust_global_fl/fedopt.md +++ b/books/fl/src/horizontal/robust_global_fl/fedopt.md @@ -8,16 +8,16 @@ Recall that modern deep learning optimizers like AdamW[^1] or AdaGrad[^2] use first- and second-order moment estimates of the stochastic gradients computed during iterative optimization to adaptively modify the model updates. At a high level, each algorithm aims to reinforce common update directions -(i.e. those with momentum) and damp update noisy directions (i.e. those with -high batch-to-batch variance). The FedOpt family[^3] of algorithms, considers -modifying the traditional FedAvg aggregation algorithm to incorporate similar -adaptations into server-side model updates in FL. +(i.e. those with momentum) and damp update elements corresponding to noisy +directions (i.e. those with high batch-to-batch variance). The FedOpt +family[^3] of algorithms, considers modifying the traditional +[FedAvg](../vanilla_fl/fedavg.md) aggregation algorithm to incorporate +similar adaptations into server-side model updates in FL. ## Mathematical motivation -In [FedAvg](../vanilla_fl/fedavg.md), recall that, after a round of local -training on each client, client model weights are combined into a single model -representation via +In FedAvg, recall that, after a round of local training on each client, +client model weights are combined into a single model representation via $$ \begin{align*} @@ -35,13 +35,13 @@ $$ \begin{align} \\mathbf{w}\_{t+1} = \\sum\_{k \\in C_t} \\frac{n_k}{n_s} \\mathbf{w}^k\_{t+1} &= \\mathbf{w}_t - \\frac{1}{C_t} \\sum\_{k \\in C_t} \\left( \\mathbf{w}_t - \\mathbf{w}^k\_{t+1} \\right), \\\\ -&= \\mathbf{w}_t - \\frac{1}{C\_t} \\sum\_{k \\in C_t} \\Delta^k\_{t+1}, \\\\ -&= \\mathbf{w}_t - \\Delta\_{t+1}. \tag{1} +&= \\mathbf{w}_t + \\frac{1}{C\_t} \\sum\_{k \\in C_t} \\Delta^k\_{t+1}, \\\\ +&= \\mathbf{w}_t + \\Delta\_{t+1}. \tag{1} \end{align} $$ -Here, \\(\\Delta^k\_{t+1} = \\mathbf{w}\_t - \\mathbf{w}^k\_{t+1}\\) is just -the vector pointing from the initial models weights and to those after local +Here, \\(\\Delta^k\_{t+1} = \\mathbf{w}^k\_{t+1} - \\mathbf{w}\_t\\) is just +the vector pointing from the initial models weights to those after local training and \\(\\Delta\_{t+1}\\) is simply the uniform average of these update vectors. @@ -53,20 +53,19 @@ equivalent to a batch-SGD update with a learning rate of 1.0 for the **server**. The "server-side" batch is the union of the batches used on each client. -The observation that \\(\Delta\_{t+1}\\) is simply a stochastic gradient -motivates treating these update directions, -\\(\\Delta^k\_{t+1} = \\mathbf{w}\_t - \\mathbf{w}^k\_{t+1}\\) like the -stochastic gradients in standard adaptive optimizers. It's important to note -that if the clients, for instance, apply multiple steps of local SGD or use -different learning rates, the exact equivalence of \\(\Delta\_{t+1}\\) to a -stochastic gradient is broken. However, it shares similarities to such a -gradient and is, therefore, called a "pseudo-gradient."[^3] +The observation that \\(-\Delta\_{t+1}\\) is simply a stochastic gradient +motivates treating these update directions like the stochastic gradients +in standard adaptive optimizers. It's important to note that if the clients, +for instance, apply multiple steps of local SGD or use different learning +rates, the exact equivalence of \\(-\Delta\_{t+1}\\) to a stochastic gradient +is broken. However, it shares similarities to such a gradient and is, +therefore, called a "pseudo-gradient."[^3] ## The algorithms: FedAdagrad, FedAdam, FedYogi Drawing inspiration from three successful, traditional adaptive optimizers, -adaptive server-side aggregation strategy FedAdaGrad, FedAdam, and FedYogi have -been proposed. See the algorithm below for details. +the adaptive server-side aggregation strategies of FedAdaGrad, FedAdam, and +FedYogi have been proposed. See the algorithm below for details.
@@ -74,7 +73,7 @@ been proposed. See the algorithm below for details.
-Those familiar with the mathematical formulations of Adagrad, Adam[^4], and +Those familiar with the mathematical formulations of Adagrad, Adam,[^4] and Yogi[^5] will recognize the general structure of these equations. Computation of \\(m_t\\), based on the average of the update directions suggested by each client through local training (\\(\Delta\_{t+1}\\)) serves to accumulate diff --git a/books/fl/src/horizontal/robust_global_fl/fedprox.md b/books/fl/src/horizontal/robust_global_fl/fedprox.md index 6cf5e33..bc783d8 100644 --- a/books/fl/src/horizontal/robust_global_fl/fedprox.md +++ b/books/fl/src/horizontal/robust_global_fl/fedprox.md @@ -12,12 +12,12 @@ first consider the kind of phenomenon that FedProx, along with other methods, attempts to counteract. To help illustrate the issue, we'll use some helpful visualizations -from researchers who proposed the SCAFFOLD method.[^2],[^3]. -Consider a two-client setting each with their own loss landscape based on their -privately held data, denoted \\(f_1\\) and \\(f_2\\). If each client has an -equal amount of data, the global loss surface, which is the loss function when -constructed from data available on both clients is equivalent to -\\((f_1 + f_2)/2\\). When performing standard federated training, the +from researchers who proposed the SCAFFOLD method.[^2],[^3] +Consider a two-client FL setting. Each client has their own loss landscape +based on their privately held data, denoted \\(f_1\\) and \\(f_2\\). If each +client has an equal amount of data, the global loss surface, which is the loss +function when constructed from all data available on both clients is equivalent +to \\((f_1 + f_2)/2\\). When performing standard federated training, the objective is to find model weights corresponding to the minimum of this global loss function. See the figure below. Note that the minima associated with the client loss functions are distinct from the global minimum. @@ -39,19 +39,20 @@ step makes steady progress towards the global minimum.
FedSGD and global convergence -
FedSGD rounds result in models making steady progress towards +
FedSGD rounds result in averaged models making steady progress towards the global minimum.
-As in the section on [FedAvg](../vanilla_fl/fedavg.md)[^4], there is a substantial -reduction in communication overhead if each client applies multiple steps of -batch SGD, optimizing the local model based on the local loss. We noted, -however, that this breaks the equivalence enjoyed, for example, by FedSGD with -centralized large-batch SGD. In settings, such as the one illustrated in the -figures thus far, with data heterogeneity and markedly different loss -landscapes this can engender various issues. One such issue is often referred -to as "client drift" and is illustrated in the figure below. +As detailed in the chapter on [FedAvg](../vanilla_fl/fedavg.md),[^4] there is a +substantial reduction in communication overhead if each client applies +multiple steps of batch SGD, optimizing the local model based on the local +loss. It was noted therein, however, that this breaks the equivalence enjoyed, +for example, by FedSGD with centralized large-batch SGD. In settings, such as +the one illustrated in the figures thus far, with data heterogeneity and +markedly different loss landscapes this can engender various issues. One such +issue is often referred to as "client drift" and is illustrated in the figure +below.
@@ -118,12 +119,13 @@ hyper-parameter tuning. Heuristically, the proximal weight may be adapted across server rounds. If the aggregated server-side training loss (average final loss on each client) fails to decrease for a round, \\(\\mu\\) is increased. If the loss improves for some number of rounds, \\(\\mu\\) is -decreased. +decreased. In the figure below, this procedure results in the fuchsia colored +line in the Figures below.
-FedProx vs. FedAvg -FedProx vs. FedAvg +FedProx vs. FedAvg +FedProx vs. FedAvg
Comparison of FedProx to FedAvg in various settings. On the top left, data is homogeneous across clients. Without adaptation FedProx struggles to out perform FedAvg. Data is heterogeneous in the other settings and FedProx performs diff --git a/books/fl/src/horizontal/robust_global_fl/moon.md b/books/fl/src/horizontal/robust_global_fl/moon.md index 44311ab..77a4eb4 100644 --- a/books/fl/src/horizontal/robust_global_fl/moon.md +++ b/books/fl/src/horizontal/robust_global_fl/moon.md @@ -4,7 +4,7 @@ {{ #aipr_header }} -The MOON[^1] algorithm is built on the same principles as the +The MOON algorithm[^1] is built on the same principles as the [FedProx](./fedprox.md)[^2] approach. That is, it targets limiting client-specific drift during local training by constraining how heavily local model updates stray from global models. The fundamental difference is the @@ -36,10 +36,11 @@ Each of which makes \\(\\ell\_{\\text{con}}(\mathbf{x};\\mathbf{w})\\) **smaller**. Contrastive loss objectives have been widely used to bring latent -representations of information closer together. For example, contrastive -learning is used extensively in CLIP[^3] as a means of pushing image-caption -text pairs closer together, while pushing unrelated image-text pairs further -apart in the CLIP model's representation space. +representations of similar inputs closer together and push dissimilar inputs +further apart. For example, contrastive learning is used extensively in +CLIP[^3] as a means of pushing image-caption text pairs closer together, while +pushing unrelated image-text pairs further apart in the CLIP model's +representation space. ## MOON models and an alternative to weight-drift penalties @@ -62,7 +63,7 @@ contrastive loss functions and their latent representations.
For server round \\(t\\), the model on the left represents the model after weight aggregation by the server. In the middle is the final model after local training on Client \\(i\\). The weights, \\(\\mathbf{w}^{t-1}\_i\\), have -been aggregated across participating clients to form \\(\\mathbf{w}\_t\\). +been aggregated across participating clients to form \\(\\mathbf{w}^t\\). Finally, the model on the right is the one being locally trained on Client \\(i\\). The output of the feature maps in these models will be used to form the local contrastive loss for Client \\(i\\). @@ -96,22 +97,23 @@ loss over all data points in the batch is computed. The idea here is similar to FedProx. In some sense, we still want to make sure that, during training, the model does not drift too -fair from the global model, as was the case in FedProx. The difference, here, +far from the global model, as was the case in FedProx. The difference, here, is that we're applying that constraint in the feature representation space, rather than directly in the model weights themselves. In the original work, MOON showed notable improvements over methods like FedProx in heterogeneous -settings. However, it does not always outperform FedProx or even FedAvg.[^4] +settings. However, it does not **always** outperform FedProx or even FedAvg.[^4] As such, there are likely scenarios where MOON is the right approach for FL in heterogeneous settings, while others might benefit from an alternative -approach. +technique. ## The algorithm The MOON algorithm is fairly similar to that of FedProx. Most server-side -aggregation strategy may be applied. However, the algorithm has some additional -memory overhead, as forward passes of three separate models must be run in -order to extract the latent representations of the data points in each training -batch. In the algorithm below, FedAvg is used as the server-side strategy. +aggregation strategies may be applied in combination with MOON. However, the +algorithm has some additional memory overhead, as forward passes of three +separate models must be run in order to extract the latent representations of +the data points in each training batch. In the algorithm below, FedAvg is used +as the server-side strategy.
From c5bda8d9859c4a57ff675dc5abdc7597a59e734b Mon Sep 17 00:00:00 2001 From: David Emerson <43939939+emersodb@users.noreply.github.com> Date: Wed, 4 Jun 2025 09:32:06 -0400 Subject: [PATCH 7/8] Addressing small PR comments. Will tackle algorithm highlight comments next --- books/fl/src/horizontal/robust_global_fl/README.md | 4 ++-- books/fl/src/horizontal/robust_global_fl/fedopt.md | 2 +- books/fl/src/horizontal/robust_global_fl/moon.md | 10 +++++----- 3 files changed, 8 insertions(+), 8 deletions(-) diff --git a/books/fl/src/horizontal/robust_global_fl/README.md b/books/fl/src/horizontal/robust_global_fl/README.md index 09cc265..5e05feb 100644 --- a/books/fl/src/horizontal/robust_global_fl/README.md +++ b/books/fl/src/horizontal/robust_global_fl/README.md @@ -34,7 +34,7 @@ model performance degrades, when the joint distributions $$ \begin{align} -P\_{\\text{train}}(X, Y) \\neq P\_{\\text{test}}(X, Y). \tag{1} +\\mathbb{P}\_{\\text{train}}(X, Y) \\neq \\mathbb{P}\_{\\text{test}}(X, Y). \tag{1} \end{align} $$ @@ -56,7 +56,7 @@ distribution, \\(\\mathbb{P}(X)\\), while the statistical relationship of labels features, \\(\\mathbb{P}(Y \\vert X)\\), remains fixed. Consider the setting of training a readmission risk model on data drawn from the patient population of a general hospital. If, for instance, that model were transferred for use at a -nearby pediatric hospital. Assuming all else equal, predictions from that model +nearby pediatric hospital, assuming all else equal, predictions from that model would be influenced by covariate drift due to the change in patient demographics. Namely, though features associated with younger patients are likely part of the general hospital population, they will, of course, be diff --git a/books/fl/src/horizontal/robust_global_fl/fedopt.md b/books/fl/src/horizontal/robust_global_fl/fedopt.md index ee1cdf7..109f77e 100644 --- a/books/fl/src/horizontal/robust_global_fl/fedopt.md +++ b/books/fl/src/horizontal/robust_global_fl/fedopt.md @@ -47,7 +47,7 @@ update vectors. Recall that, if each client uses a fixed learning rate, \\(\eta\\), and performs a single, full gradient update, FedAvg is equivalent to centralized -large-batch SGD. Similarly, in this case, if each client performs on step of +large-batch SGD. Similarly, in this case, if each client performs one step of batch SGD with a learning rate of 1.0, then the update in Equation (1) is equivalent to a batch-SGD update with a learning rate of 1.0 for the **server**. The "server-side" batch is the union of the batches used on each diff --git a/books/fl/src/horizontal/robust_global_fl/moon.md b/books/fl/src/horizontal/robust_global_fl/moon.md index 77a4eb4..9399740 100644 --- a/books/fl/src/horizontal/robust_global_fl/moon.md +++ b/books/fl/src/horizontal/robust_global_fl/moon.md @@ -106,14 +106,14 @@ As such, there are likely scenarios where MOON is the right approach for FL in heterogeneous settings, while others might benefit from an alternative technique. -## The algorithm +## The Algorithm The MOON algorithm is fairly similar to that of FedProx. Most server-side aggregation strategies may be applied in combination with MOON. However, the -algorithm has some additional memory overhead, as forward passes of three -separate models must be run in order to extract the latent representations of -the data points in each training batch. In the algorithm below, FedAvg is used -as the server-side strategy. +algorithm has some additional compute and memory overhead, as forward passes +of three separate models must be run in order to extract the latent +representations of the data points in each training batch. In the algorithm +below, FedAvg is used as the server-side strategy.
From f86f387b7594e285c2f3d2e939ed23f334da4ed1 Mon Sep 17 00:00:00 2001 From: David Emerson <43939939+emersodb@users.noreply.github.com> Date: Wed, 4 Jun 2025 11:06:16 -0400 Subject: [PATCH 8/8] Adding requested highlighting to the moon and fedprox algorithms. Also switching out fedsgd algorithm to not have Algorithm 2 --- books/fl/src/assets/algorithm-fedprox.svg | 1252 +++++++++--------- books/fl/src/assets/algorithm-fedsgd.svg | 447 +++++++ books/fl/src/assets/algorithm-moon.svg | 256 ++-- books/fl/src/horizontal/vanilla_fl/fedsgd.md | 2 +- 4 files changed, 1210 insertions(+), 747 deletions(-) create mode 100644 books/fl/src/assets/algorithm-fedsgd.svg diff --git a/books/fl/src/assets/algorithm-fedprox.svg b/books/fl/src/assets/algorithm-fedprox.svg index bd40b18..e1a8956 100644 --- a/books/fl/src/assets/algorithm-fedprox.svg +++ b/books/fl/src/assets/algorithm-fedprox.svg @@ -5,9 +5,9 @@ version="1.1" id="svg1" width="458.28131" - height="313.67999" - viewBox="0 0 458.28131 313.67999" - sodipodi:docname="algorithm-fedprox.svg" + height="321.504" + viewBox="0 0 458.28131 321.504" + sodipodi:docname="fedprox.svg" inkscape:version="1.4.2 (ebf0e940, 2025-05-08)" xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape" xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd" @@ -22,644 +22,640 @@ inkscape:pageopacity="0.0" inkscape:pagecheckerboard="0" inkscape:deskcolor="#d1d1d1" - 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diff --git a/books/fl/src/assets/algorithm-fedsgd.svg b/books/fl/src/assets/algorithm-fedsgd.svg new file mode 100644 index 0000000..6dbc801 --- /dev/null +++ b/books/fl/src/assets/algorithm-fedsgd.svg @@ -0,0 +1,447 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/books/fl/src/assets/algorithm-moon.svg b/books/fl/src/assets/algorithm-moon.svg index 6bcbd8c..5a8be95 100644 --- a/books/fl/src/assets/algorithm-moon.svg +++ b/books/fl/src/assets/algorithm-moon.svg @@ -5,9 +5,9 @@ version="1.1" id="svg1" width="458.28131" - height="310.72266" - viewBox="0 0 458.28131 310.72266" - sodipodi:docname="output.svg" + height="324.944" + viewBox="0 0 458.28131 324.944" + sodipodi:docname="moon.svg" inkscape:version="1.4.2 (ebf0e940, 2025-05-08)" xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape" xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd" @@ -22,13 +22,13 @@ inkscape:pageopacity="0.0" inkscape:pagecheckerboard="0" inkscape:deskcolor="#d1d1d1" - inkscape:zoom="0.76710396" - inkscape:cx="220.96093" - inkscape:cy="149.26269" - inkscape:window-width="1920" - inkscape:window-height="948" - inkscape:window-x="1512" - inkscape:window-y="25" + inkscape:zoom="0.79808475" + inkscape:cx="193.58846" + inkscape:cy="229.29896" + inkscape:window-width="1512" + inkscape:window-height="837" + inkscape:window-x="0" + inkscape:window-y="38" inkscape:window-maximized="1" inkscape:current-layer="g1" /> + transform="matrix(1.3333333,0,0,-1.3333333,205.19067,699.32933)" /> + transform="matrix(1.3333333,0,0,-1.3333333,225.64667,651.86267)" /> + transform="matrix(1.3333333,0,0,-1.3333333,246.104,604.03867)" /> + transform="matrix(1.3333333,0,0,-1.3333333,266.56,586.77067)" /> + - + id="g72" + aria-label="Generatez,z glob ,z prev foreachx∈b." + transform="matrix(1.3333333,0,0,1.3333333,284.09467,555.34533)"> - + + + + + - - - + + + + + + transform="matrix(1.3333333,0,0,-1.3333333,178.35733,719.78667)" /> diff --git a/books/fl/src/horizontal/vanilla_fl/fedsgd.md b/books/fl/src/horizontal/vanilla_fl/fedsgd.md index 40dd0f5..c33bb55 100644 --- a/books/fl/src/horizontal/vanilla_fl/fedsgd.md +++ b/books/fl/src/horizontal/vanilla_fl/fedsgd.md @@ -118,7 +118,7 @@ client receives the final model as described by the weights \\(\mathbf{w}\_T\\).
-FedSGD Algorithm +FedSGD Algorithm
-Local Model 1 -Local Model 2 +Local Model 1 +Local Model 2
Two clients with different datasets. Note that each holds a slightly different view of the feature space. Notably, Client 1 (left) has a distinct cluster of data points in the bottom right and fewer points labeled in @@ -127,8 +127,8 @@ distributions from a global perspective.
-FedAvg Model -FedDF Model +FedAvg Model +FedDF Model
Model resulting from FedAvg (left) compared with the model trained using FedDF[^2] (right).
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b/books/fl/src/assets/algorithm-fedprox.svg @@ -0,0 +1,665 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +. +end +end +Sendw +k +toserverforaggregation. +end +w +t+1 += +P +k∈C +t +n +k" + transform="matrix(1.3333333,0,0,1.3333333,280.10933,585.972)"> + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/books/fl/src/assets/combined_loss_objective.svg b/books/fl/src/assets/combined_loss_objective.svg new file mode 100644 index 0000000..c427a50 --- /dev/null +++ b/books/fl/src/assets/combined_loss_objective.svg @@ -0,0 +1,223 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/books/fl/src/assets/fedavg_drift.svg b/books/fl/src/assets/fedavg_drift.svg new file mode 100644 index 0000000..a275ebc --- /dev/null +++ b/books/fl/src/assets/fedavg_drift.svg @@ -0,0 +1,881 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/books/fl/src/assets/fedsgd_steps.svg b/books/fl/src/assets/fedsgd_steps.svg new file mode 100644 index 0000000..b36a495 --- /dev/null +++ b/books/fl/src/assets/fedsgd_steps.svg @@ -0,0 +1,994 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/books/fl/src/horizontal/robust_global_fl/fedprox.md b/books/fl/src/horizontal/robust_global_fl/fedprox.md index 6f4e40e..d75a7ee 100644 --- a/books/fl/src/horizontal/robust_global_fl/fedprox.md +++ b/books/fl/src/horizontal/robust_global_fl/fedprox.md @@ -1 +1,154 @@ + + # FedProx + +{{ #aipr_header }} + +The FedProx algorithm[^1] is one of the earliest approaches specifically aimed +at addressing the optimization challenges associated with data heterogeneity in +FL. At its core, the FedProx algorithm is quite straightforward. However, +prior to diving into the modifications proposed in the FedProx approach, we'll +first consider the kind of phenomenon that FedProx, along with other methods, +attempts to counteract. + +To help illustrate the issue, we'll use some helpful visualizations +from researchers who proposed the SCAFFOLD method.[^2],[^3]. +Consider a two-client setting each with their own loss landscape based on their +privately held data, denoted \\(f_1\\) and \\(f_2\\). If each client has an +equal amount of data, the global loss surface, which is the loss function when +constructed from data available on both clients is equivalent to +\\((f_1 + f_2)/2\\). When performing standard federated training, the +objective is to find model weights corresponding to the minimum of this global +loss function. See the figure below. Note that the minima associated with the +client loss functions are distinct from the global minimum. + +
+
+Combined loss objective +
Comparison of local loss landscapes for two clients with the combined global loss.
+
+
+ +Recall that optimization with [FedSGD](../vanilla_fl/fedsgd.md) is equivalent +to centralized large-batch SGD. That is, rounds of FedSGD are equivalent to +optimizing the global loss function, expressed here as \\((f_1 + f_2)/2\\). As +such, with a properly tuned learning rate, FedSGD will converge to the +global optimum, as illustrated in the figure below. Each averaged gradient +step makes steady progress towards the global minimum. + +
+
+FedSGD and global convergence +
FedSGD rounds result in models making steady progress towards +the global minimum.
+
+
+ +As in the section on [FedAvg](../vanilla_fl/fedavg.md), there is a substantial +reduction in communication overhead if each client applies multiple steps of +batch SGD, optimizing the local model based on the local loss. We noted, +however, that this breaks the equivalence enjoyed, for example, by FedSGD with +centralized large-batch SGD. In settings, such as the one illustrated in the +figures thus far, with data heterogeneity and markedly different loss +landscapes this can engender various issues. One such issue is often referred +to as "client drift" and is illustrated in the figure below. + +
+
+FedAvg and the influence of local drift +
Illustration of "client drift" in FedAvg updates caused by +differences in the shape of the local loss functions of each client.
+
+
+ +In the figure, each client is applying three local steps of batch SGD before +sending the resulting weights to server for aggregation. The grey dots +represent the updates using FedSGD for three rounds from the previous figure. +The update using FedAvg deviates quite a bit from this path with a distinct +drift towards the minima of Client 2. Drifts of this kind can be induced by the +shape of the local loss surface and cause issues with FedAvg, such as slowed +convergence. + +## The Math + +The general idea for FedProx is to limit models from drifting too far during +local training. For a server round \\(t\\), consider the aggregated weights +\\(\\mathbf{w}\_t\\). For a given client \\(k\\), let +\\(\\ell_k(b; \\mathbf{w})\\) denote the local loss function for a batch, +\\(b\\), of data, parameterized by model weights \\(\\mathbf{w}\\). The primary +modification of FedAvg in the FedProx algorithm is to augment +\\(\\ell_k(b; \\mathbf{w})\\) with a penalty term such that, for \\(\mu > 0\\), +the local loss becomes + +$$ +\begin{align*} +\\ell_k(b; \\mathbf{w}) + \\mu \\Vert \\mathbf{w} - \\mathbf{w_t} \\Vert^2. +\end{align*} +$$ + +The penalty term is referred to as the proximal loss. It penalizes significant +deviation from the global model during local training such that loss +optimization must trade off improvements in the standard loss with potential +divergence from the original model weights. Revisiting the loss surfaces above, +the FedProx penalty term alters the loss surface to make client drift less +attractive, unless it leads to significant performance gains. + +## The Algorithm + +The FedProx algorithm is very similar to that of FedAvg, with the only +modification coming in the local update calculations. + +
+
+FedProx Algorithms +
+
+ +## Adapting \\(\\mu\\) + +For a well-tuned \\(\\mu\\), FedProx has been shown to outperform FedAvg under +heterogeneous data conditions. It is widely applied, both because it is a +simple modification to the FedAvg framework and because it works fairly well +across a number of tasks. However, in settings where the data is homogeneous, +FedProx has been shown to under-perform compared to FedAvg when \\(\\mu>0\\). +See the top left of the figure below. + +Because of this, the authors of FedProx offer an alternative to extensive +hyper-parameter tuning. Heuristically, the proximal weight may be adapted +across server rounds. If the aggregated server-side training loss +(average final loss on each client) fails to decrease for a round, \\(\\mu\\) +is increased. If the loss improves for some number of rounds, \\(\\mu\\) is +decreased. + +
+
+FedProx vs. FedAvg +FedProx vs. FedAvg +
Comparison of FedProx to FedAvg in various settings. On the top left, +data is homogeneous across clients. Without adaptation FedProx struggles to out +perform FedAvg. Data is heterogeneous in the other settings and FedProx performs +well with and without adaptation. +
+
+
+ +#### References & Useful Links + +[^1]: + [T. Li, A. K. Sahu, M. Zaheer, M. Sanjabi, A. Talwalkar, and V. Smith. + Federated optimization in heterogeneous networks. In I. Dhillon, + D. Papailiopoulos, and V. Sze, editors, Proceedings of Machine Learning and + Systems, volume 2, pages 429–450, 2020.](https://arxiv.org/pdf/1812.06127) + +[^2]: + [S. P. Karimireddy, S. Kale, M. Mohri, S. Reddi, S. Stich, and A. T. Suresh. + SCAFFOLD: Stochastic controlled averaging for federated learning. In + Hal Daumé III and Aarti Singh, editors, Proceedings of the 37th + International Conference on Machine Learning, volume 119 of Proceedings of + Machine Learning Research, pages 5132–5143. PMLR, 13–18 Jul 2020.](https://www.microsoft.com/en-us/research/publication/frustratingly-easy-neural-domain-adaptation/) + +[^3]: + [Images adapted from Talk on "Stochastic Controlled Averaging for + Federated Learning"](https://docs.google.com/presentation/d/1SYhRC6NMEMJJL2FTGDu5DJeINJBkbwFVGd2Emy43fk0/edit) + +{{#author emersodb}} From 904ae501a0c6649429a9721c253ad9ece20f12a2 Mon Sep 17 00:00:00 2001 From: David Emerson <43939939+emersodb@users.noreply.github.com> Date: Fri, 16 May 2025 17:02:06 -0400 Subject: [PATCH 5/8] Moon pocket reference --- books/fl/src/assets/SplitModels.svg | 4 + books/fl/src/assets/algorithm-moon.svg | 627 ++++++++++++++++++ .../horizontal/robust_global_fl/fedprox.md | 7 +- .../src/horizontal/robust_global_fl/moon.md | 150 ++++- 4 files changed, 786 insertions(+), 2 deletions(-) create mode 100644 books/fl/src/assets/SplitModels.svg create mode 100644 books/fl/src/assets/algorithm-moon.svg diff --git a/books/fl/src/assets/SplitModels.svg b/books/fl/src/assets/SplitModels.svg new file mode 100644 index 0000000..d0d4588 --- /dev/null +++ b/books/fl/src/assets/SplitModels.svg @@ -0,0 +1,4 @@ + + + +
Feature Map
Input Features
Classifier
Label
Global Model
Feature Map
Input Features
Classifier
Label
Previous Local Model
Feature Map
Input Features
Classifier
Label
Current Local Model
diff --git a/books/fl/src/assets/algorithm-moon.svg b/books/fl/src/assets/algorithm-moon.svg new file mode 100644 index 0000000..6bcbd8c --- /dev/null +++ b/books/fl/src/assets/algorithm-moon.svg @@ -0,0 +1,627 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/books/fl/src/horizontal/robust_global_fl/fedprox.md b/books/fl/src/horizontal/robust_global_fl/fedprox.md index d75a7ee..6cf5e33 100644 --- a/books/fl/src/horizontal/robust_global_fl/fedprox.md +++ b/books/fl/src/horizontal/robust_global_fl/fedprox.md @@ -44,7 +44,7 @@ the global minimum.