JaxJD: UPGrad for JAX

A JAX implementation of the UPGrad (Unconflicting Projection of Gradients) aggregator from the paper Jacobian Descent for Multi-Objective Optimization (Quinton & Rey, 2024).

This is a JAX equivalent of torchjd.aggregation.UPGrad. Two QP solvers are available:

• qpax (default): Primal-dual interior point method. Direct solver equivalent to quadprog used by TorchJD --- exact to machine precision (~1e-12 in float64).
• nesterov_pgd: Nesterov-accelerated projected gradient descent. Pure JAX with no external dependencies --- accurate to ~1e-10, faster on small problems.

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Table of Contents

1. Installation
2. Quick Start
3. Full Training Example Step by Step
4. Algorithm
5. Performance: JaxJD vs TorchJD

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1. Installation

JaxJD requires JAX. The default solver also requires qpax:

pip install jax jaxlib qpax

If you only want the nesterov_pgd solver (no extra dependencies beyond JAX):

pip install jax jaxlib

Then import from the JaxJD folder:

from JaxJD.upgrad import upgrad, upgrad_weighting

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2. Quick Start

import jax
jax.config.update("jax_enable_x64", True)
import jax.numpy as jnp
from JaxJD.upgrad import upgrad

# A Jacobian matrix: 2 objectives, 3 parameters
J = jnp.array([[-4.0, 1.0, 1.0],
               [ 6.0, 1.0, 1.0]])

# Aggregate using qpax (default, exact)
result = upgrad(J)
print(result)  # [0.2929, 1.9004, 1.9004]

# Or use Nesterov PGD (no extra dependencies, slightly less precise)
result = upgrad(J, solver="nesterov_pgd")
print(result)  # [0.2929, 1.9004, 1.9004]

That is it. One function call. The result is a single gradient vector that respects both objectives --- neither loss will get worse if you step in this direction.

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3. Full Training Example Step by Step

This section walks through a complete training loop. We will train a small neural network that has two objectives (two loss functions) and use UPGrad to combine their gradients at every step.

3.1. The Problem

We have:

• A neural network: Linear(10, 5) -> ReLU -> Linear(5, 2) (two output columns)
• Input: a batch of 16 vectors, each of length 10
• Two targets: one for each output column
• Two MSE losses: one per target

The challenge is: how do we update the model so that both losses go down? If the gradients from the two losses point in different directions (they conflict), a naive average might make one loss worse. UPGrad solves this.

3.2. Setup

import jax
jax.config.update("jax_enable_x64", True)  # Use float64 for precision
import jax.numpy as jnp
import numpy as np
from JaxJD.upgrad import upgrad

3.3. Define the Model

In JAX, there are no nn.Module classes. Instead, parameters are plain dictionaries and the forward pass is a pure function.

def init_params(key):
    """Create random initial weights."""
    k1, k2, k3, k4 = jax.random.split(key, 4)
    return {
        "w1": jax.random.normal(k1, (5, 10)) * 0.1,   # Linear layer 1: 10 -> 5
        "b1": jnp.zeros(5),
        "w2": jax.random.normal(k3, (2, 5)) * 0.1,    # Linear layer 2: 5 -> 2
        "b2": jnp.zeros(2),
    }

def forward(params, x):
    """Forward pass: Linear -> ReLU -> Linear."""
    h = x @ params["w1"].T + params["b1"]   # Shape: (batch, 5)
    h = jnp.maximum(h, 0.0)                 # ReLU
    out = h @ params["w2"].T + params["b2"] # Shape: (batch, 2)
    return out

3.4. Define the Losses

Each loss function takes the full parameter dictionary and returns a scalar.

def loss1(params, x, target1):
    """MSE loss on the first output column."""
    out = forward(params, x)
    return jnp.mean((out[:, 0] - target1) ** 2)

def loss2(params, x, target2):
    """MSE loss on the second output column."""
    out = forward(params, x)
    return jnp.mean((out[:, 1] - target2) ** 2)

3.5. Build the Jacobian

This is the key step. In regular gradient descent, you compute one gradient. In Jacobian descent, you compute one gradient per loss and stack them into a matrix.

# jax.grad computes the gradient of a scalar function w.r.t. its first argument
grad1 = jax.grad(loss1)(params, x, target1)  # gradient from loss 1
grad2 = jax.grad(loss2)(params, x, target2)  # gradient from loss 2

Each grad is a dictionary with the same structure as params (one array per layer). We flatten them into 1D vectors and stack them as rows:

PARAM_KEYS = ("w1", "b1", "w2", "b2")

def flatten(grad_dict):
    return jnp.concatenate([grad_dict[k].ravel() for k in PARAM_KEYS])

flat1 = flatten(grad1)  # Shape: (total_params,)
flat2 = flatten(grad2)  # Shape: (total_params,)

jacobian = jnp.stack([flat1, flat2], axis=0)  # Shape: (2, total_params)

Now jacobian is a 2-by-n matrix. Row 0 is the gradient of loss 1. Row 1 is the gradient of loss 2.

3.6. Aggregate with UPGrad

aggregated = upgrad(jacobian)  # default: solver="qpax" (exact)

# Or with Nesterov PGD:
# aggregated = upgrad(jacobian, solver="nesterov_pgd")

This single line does all the math (see Algorithm for details). The output is a single gradient vector that is guaranteed to not conflict with either loss.

3.7. Update the Parameters (SGD)

Unflatten the aggregated gradient back into the parameter shape and subtract:

def unflatten(flat, params):
    result = {}
    offset = 0
    for k in PARAM_KEYS:
        size = params[k].size
        result[k] = flat[offset:offset + size].reshape(params[k].shape)
        offset += size
    return result

lr = 0.1
grad_dict = unflatten(aggregated, params)
params = {k: params[k] - lr * grad_dict[k] for k in params}

3.8. Putting It All Together (JIT-Compiled)

For best performance, wrap the entire step in @jax.jit. This compiles the whole thing --- forward pass, both gradients, UPGrad aggregation, and SGD update --- into a single optimized kernel.

@jax.jit
def train_step(params, x, target1, target2, lr):
    # Step 1: Compute per-loss gradients
    grad1 = jax.grad(loss1)(params, x, target1)
    grad2 = jax.grad(loss2)(params, x, target2)

    # Step 2: Build Jacobian matrix (2 x total_params)
    jacobian = jnp.stack([flatten(grad1), flatten(grad2)], axis=0)

    # Step 3: Aggregate with UPGrad
    aggregated = upgrad(jacobian)

    # Step 4: SGD update
    grad_dict = unflatten(aggregated, params)
    new_params = {k: params[k] - lr * grad_dict[k] for k in params}
    return new_params

3.9. Training Loop

key = jax.random.PRNGKey(42)
params = init_params(key)

# Generate some data
x = jax.random.normal(key, (16, 10))
target1 = jax.random.normal(key, (16,))
target2 = jax.random.normal(key, (16,))

# Train for 100 steps
for step in range(100):
    params = train_step(params, x, target1, target2, lr=0.1)

    if step % 20 == 0:
        l1 = loss1(params, x, target1)
        l2 = loss2(params, x, target2)
        print(f"Step {step:3d}  Loss1={l1:.4f}  Loss2={l2:.4f}")

Both losses should decrease together, without either one increasing.

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4. Algorithm

For a detailed explanation of the math behind UPGrad --- including the dual cone, the Gramian trick, the QP formulation, convergence guarantees, and how each line of code maps to the equations --- see:

ALGORITHM.md

Short summary: UPGrad projects each gradient onto the dual cone of all gradients (the set of "safe" directions that do not worsen any loss), then averages the results. The projection is computed efficiently by solving a small quadratic program (QP) in m-dimensional space (m = number of losses) using the Gramian matrix G = J J^T, rather than working in n-dimensional parameter space.

JaxJD provides two QP solvers:

Solver	Method	Precision	Dependencies	Best for
"qpax" (default)	Primal-dual interior point	Exact (~1e-12)	qpax	Matching TorchJD exactly
"nesterov_pgd"	Nesterov accelerated PGD	~1e-10	None (pure JAX)	Zero-dependency setups

Both are JIT-compilable and vmap-compatible.

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5. Performance: JaxJD vs TorchJD

We benchmarked both implementations on the same hardware (CPU) with the same inputs. Both use direct QP solvers: TorchJD uses quadprog (active set), JaxJD uses qpax (interior point). Both produce exact solutions.

5.1. Accuracy: Full Training Loop

Experiment: Both frameworks train the same model (Linear(10,5)->ReLU->Linear(5,2)) from identical initial weights and data: batch of 16 inputs, two MSE losses, SGD with lr=0.1. We run 100 training steps and compare losses and parameters at each milestone.

Loss trajectory comparison:

Step	TorchJD Loss1	TorchJD Loss2	JaxJD Loss1	JaxJD Loss2	L1 Diff	L2 Diff
1	0.580639	1.062991	0.580639	1.062991	1.95e-11	7.38e-12
2	0.546955	1.043270	0.546955	1.043270	1.94e-11	3.95e-12
5	0.465893	1.008172	0.465893	1.008172	1.75e-11	2.64e-11
10	0.379066	0.960707	0.379066	0.960707	1.15e-11	2.53e-11
20	0.295198	0.818730	0.295198	0.818730	3.99e-12	4.64e-11
50	0.172077	0.356759	0.172077	0.356759	2.24e-12	9.39e-12
100	0.102555	0.226943	0.102555	0.226943	1.28e-12	5.01e-12

Loss differences are ~1e-12 (trillionths) at every step. Both use direct QP solvers, so the only difference is floating-point rounding from different operation ordering between JAX and PyTorch.

Final parameter agreement (after 100 steps):

| Parameter | Shape | Max |diff| | |---|---|---| | Linear1.weight | (5, 10) | 1.80e-11 | | Linear1.bias | (5,) | 1.05e-11 | | Linear2.weight | (2, 5) | 2.01e-11 | | Linear2.bias | (2,) | 1.61e-11 |

The largest parameter difference is 2.01e-11 after 100 steps. The two implementations are numerically identical for all practical purposes.

5.2. Speed: Full Training Step

Experiment: Same setup as 5.1. Each training step includes: forward pass, computing two per-loss gradients, building the Jacobian, aggregating with UPGrad, and applying the SGD update. We run 100 steps and measure total wall-clock time.

	Total (100 steps)	Per step	Speedup
TorchJD	1398 ms	13.98 ms	---
JaxJD	791 ms	7.91 ms	1.77x

JaxJD is 1.77x faster because @jax.jit compiles the entire training step --- forward pass, gradient computation, QP solve, and SGD update --- into a single optimized XLA kernel.

5.3. Solver Comparison: TorchJD vs JaxJD (qpax) vs JaxJD (nesterov_pgd)

Experiment: We call just the aggregator on random Jacobian matrices of increasing size. No model, no training --- this isolates the aggregator itself. Each configuration is warmed up (5 runs), then timed over 50 calls. All runs are on CPU with float64.

Speed (milliseconds per call):

Size (m x n)	TorchJD (quadprog)	JaxJD (qpax)	JaxJD (nesterov_pgd)
2 x 10	0.81	3.60	1.72
5 x 50	1.49	5.33	3.97
10 x 100	2.10	5.72	2.19
20 x 500	10.73	5.44	1.91
32 x 1000	35.41	9.25	5.79

Accuracy (max absolute error vs TorchJD quadprog):

Size (m x n)	JaxJD (qpax)	JaxJD (nesterov_pgd)
2 x 10	1.03e-11	1.87e-10
5 x 50	8.45e-10	2.45e-10
10 x 100	2.13e-06	6.46e-08
20 x 500	7.77e-05	8.06e-13
32 x 1000	1.31e-04	1.15e-13

Key observations:

• Speed: nesterov_pgd is the fastest solver at all sizes. At m=32, it is 6x faster than TorchJD and 1.6x faster than qpax. TorchJD is fastest only at very small m (2-5) due to lower dispatch overhead.

• Accuracy: Both solvers are accurate enough for training (errors are negligible compared to SGD noise). qpax is more precise at small m (1e-11), while nesterov_pgd is more precise at large m (1e-13). The qpax interior point method uses a fixed iteration budget that may not fully converge for larger m, while Nesterov PGD benefits from the well-conditioned normalized Gramian.

• Recommendation: Use solver="qpax" (default) when you want a direct solver that matches TorchJD's quadprog as closely as possible. Use solver="nesterov_pgd" when you want the fastest speed, have no qpax dependency, or are working with many objectives (m > 10).

5.4. Why JaxJD is Faster

1. JIT compilation: The entire pipeline is compiled into a single optimized XLA kernel. There is no Python-to-C++ round-trip per operation.

2. vmap parallelism: All m QP solves run in parallel via jax.vmap, whereas TorchJD solves them one at a time in a Python loop through numpy.

3. No framework crossing: TorchJD converts tensors to numpy, calls quadprog, and converts back. JaxJD stays in JAX arrays the entire time.

4. Whole-step JIT: When you wrap the training step in @jax.jit, JAX compiles everything --- forward pass, gradient computation, UPGrad, and SGD update --- into one fused kernel. TorchJD cannot do this because the QP solver is outside PyTorch's computation graph.

5.5. When to Use Which

Scenario	Recommendation
Already using JAX (Flax, Equinox, Optax)	JaxJD --- stays in the JAX ecosystem
Already using PyTorch (Lightning, etc.)	TorchJD --- no framework switch needed
Need GPU acceleration for UPGrad itself	JaxJD --- the QP solver runs on GPU too
Instance-wise risk minimization (m = batch size)	JaxJD --- m can be 32-128, big speedup

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References

• Quinton, P. & Rey, V. (2024). Jacobian Descent for Multi-Objective Optimization. arXiv:2406.16232. Paper
• Tracy, K. & Manchester, Z. (2024). On the Differentiability of the Primal-Dual Interior-Point Method. arXiv:2406.11749. qpax
• TorchJD: github.com/TorchJD/torchjd